| L(s) = 1 | + 4·3-s − 4·5-s + 6·7-s + 11·9-s + 8·11-s + 12·13-s − 16·15-s − 8·17-s + 24·21-s − 4·23-s + 2·25-s + 20·27-s − 18·29-s + 16·31-s + 32·33-s − 24·35-s − 6·37-s + 48·39-s − 4·41-s − 28·43-s − 44·45-s + 11·49-s − 32·51-s − 20·53-s − 32·55-s + 24·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 2.26·7-s + 11/3·9-s + 2.41·11-s + 3.32·13-s − 4.13·15-s − 1.94·17-s + 5.23·21-s − 0.834·23-s + 2/5·25-s + 3.84·27-s − 3.34·29-s + 2.87·31-s + 5.57·33-s − 4.05·35-s − 0.986·37-s + 7.68·39-s − 0.624·41-s − 4.26·43-s − 6.55·45-s + 11/7·49-s − 4.48·51-s − 2.74·53-s − 4.31·55-s + 3.12·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.254414717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.254414717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 25 T^{2} - 78 T^{3} + 204 T^{4} - 78 p T^{5} + 25 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 8 p T^{3} - 52 p T^{4} + 8 p^{2} T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 164 T^{3} + 628 T^{4} + 164 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 96 T^{3} - 124 T^{4} + 96 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 244 T^{3} + 1588 T^{4} + 244 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 18 T + 177 T^{2} + 1242 T^{3} + 7052 T^{4} + 1242 p T^{5} + 177 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 912 T^{3} + 5822 T^{4} - 912 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T + 85 T^{2} + 438 T^{3} + 4404 T^{4} + 438 p T^{5} + 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 424 T^{3} + 2092 T^{4} + 424 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 28 T + 365 T^{2} + 3060 T^{3} + 20876 T^{4} + 3060 p T^{5} + 365 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8006 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1580 T^{3} + 11806 T^{4} + 1580 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 153 T^{2} + 1176 T^{3} - 21292 T^{4} + 1176 p T^{5} + 153 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 121 T^{2} + 1242 T^{3} + 15012 T^{4} + 1242 p T^{5} + 121 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 272 T^{3} - 1748 T^{4} - 272 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 16134 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 28 T + 197 T^{2} + 2408 T^{3} - 48788 T^{4} + 2408 p T^{5} + 197 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 179 T^{2} - 22 T^{3} + 23692 T^{4} - 22 p T^{5} - 179 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.610346158726420824751441664746, −8.252047313603479156648744273117, −8.230612765454673446587582806981, −8.177037954657857910241439387424, −8.107498813846741626312997285450, −7.59310500925393735859461081877, −7.31063719442197265825231105331, −6.90604809138374494331939667463, −6.74971256443723792136709754852, −6.64788167196521563928874840225, −6.15443319531702183703291868572, −5.81953960837875859378702042148, −5.58570420194405543104306933732, −4.67307418932288614448509143386, −4.58193133376093330025124441029, −4.35584187622343341792790672418, −4.34664307617958722339205963536, −3.64456328275796076045641318957, −3.57335143420865868677238473167, −3.39786197162718606891303099303, −3.34057032696346975032708691445, −1.90331966258010760259918966033, −1.83917472684112210544127837166, −1.83275537085266612089379124998, −1.25771307979135068966093774660,
1.25771307979135068966093774660, 1.83275537085266612089379124998, 1.83917472684112210544127837166, 1.90331966258010760259918966033, 3.34057032696346975032708691445, 3.39786197162718606891303099303, 3.57335143420865868677238473167, 3.64456328275796076045641318957, 4.34664307617958722339205963536, 4.35584187622343341792790672418, 4.58193133376093330025124441029, 4.67307418932288614448509143386, 5.58570420194405543104306933732, 5.81953960837875859378702042148, 6.15443319531702183703291868572, 6.64788167196521563928874840225, 6.74971256443723792136709754852, 6.90604809138374494331939667463, 7.31063719442197265825231105331, 7.59310500925393735859461081877, 8.107498813846741626312997285450, 8.177037954657857910241439387424, 8.230612765454673446587582806981, 8.252047313603479156648744273117, 8.610346158726420824751441664746
