| L(s) = 1 | + 4·3-s + 8·5-s − 4·7-s + 10·9-s + 4·11-s + 32·15-s + 12·19-s − 16·21-s − 12·23-s + 30·25-s + 20·27-s + 8·29-s − 16·31-s + 16·33-s − 32·35-s + 12·41-s + 12·43-s + 80·45-s + 8·49-s + 16·53-s + 32·55-s + 48·57-s − 4·59-s − 40·63-s + 12·67-s − 48·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.57·5-s − 1.51·7-s + 10/3·9-s + 1.20·11-s + 8.26·15-s + 2.75·19-s − 3.49·21-s − 2.50·23-s + 6·25-s + 3.84·27-s + 1.48·29-s − 2.87·31-s + 2.78·33-s − 5.40·35-s + 1.87·41-s + 1.82·43-s + 11.9·45-s + 8/7·49-s + 2.19·53-s + 4.31·55-s + 6.35·57-s − 0.520·59-s − 5.03·63-s + 1.46·67-s − 5.77·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.24038468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.24038468\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T^{2} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 48 T^{3} + 2 T^{4} + 48 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 492 T^{3} + 2402 T^{4} - 492 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 34 T^{2} - 200 T^{3} + 1026 T^{4} - 200 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 144 T^{3} + 2 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} + 564 T^{3} - 6334 T^{4} + 564 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} + 128 T^{3} - 3294 T^{4} + 128 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 668 T^{3} - 2718 T^{4} - 668 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 240 T^{3} + 2 T^{4} + 240 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 876 T^{3} + 7010 T^{4} - 876 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 1052 T^{3} - 4254 T^{4} + 1052 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.047305660178889925435735199499, −7.39382802977332652849392586754, −7.38420758661995690699161603558, −7.16783059579153792762370017628, −7.09312695549494287295714707077, −6.60508396574732304213570225049, −6.28314833168372276568282375420, −6.24037218533497487655604270079, −5.94169811290984680695497216588, −5.64845363974742326484798137916, −5.53664352247656033853122813304, −5.28722972975958068411644569702, −5.07199755704602448152340340110, −4.27505666431580358242657037588, −4.04639799900959809213827091605, −3.91543387539799411328306867754, −3.79358978924234690602699085188, −3.14384147017070970659978491641, −2.99972411211255227672338869200, −2.48547718427633467974134033716, −2.45425989145403632854503248510, −2.33073880005159011949646015232, −1.58404581371373945712015655848, −1.44977343637243623639026218054, −1.15968335931606288035699992566,
1.15968335931606288035699992566, 1.44977343637243623639026218054, 1.58404581371373945712015655848, 2.33073880005159011949646015232, 2.45425989145403632854503248510, 2.48547718427633467974134033716, 2.99972411211255227672338869200, 3.14384147017070970659978491641, 3.79358978924234690602699085188, 3.91543387539799411328306867754, 4.04639799900959809213827091605, 4.27505666431580358242657037588, 5.07199755704602448152340340110, 5.28722972975958068411644569702, 5.53664352247656033853122813304, 5.64845363974742326484798137916, 5.94169811290984680695497216588, 6.24037218533497487655604270079, 6.28314833168372276568282375420, 6.60508396574732304213570225049, 7.09312695549494287295714707077, 7.16783059579153792762370017628, 7.38420758661995690699161603558, 7.39382802977332652849392586754, 8.047305660178889925435735199499
