| L(s) = 1 | + 6·5-s + 16·7-s + 34·11-s + 24·13-s − 8·17-s + 46·23-s + 11·25-s + 50·31-s + 96·35-s + 18·37-s − 146·41-s − 78·43-s + 90·47-s + 128·49-s − 16·53-s + 204·55-s − 192·61-s + 144·65-s − 134·67-s + 50·71-s + 80·73-s + 544·77-s − 150·83-s − 48·85-s + 384·91-s − 186·97-s + 386·101-s + ⋯ |
| L(s) = 1 | + 6/5·5-s + 16/7·7-s + 3.09·11-s + 1.84·13-s − 0.470·17-s + 2·23-s + 0.439·25-s + 1.61·31-s + 2.74·35-s + 0.486·37-s − 3.56·41-s − 1.81·43-s + 1.91·47-s + 2.61·49-s − 0.301·53-s + 3.70·55-s − 3.14·61-s + 2.21·65-s − 2·67-s + 0.704·71-s + 1.09·73-s + 7.06·77-s − 1.80·83-s − 0.564·85-s + 4.21·91-s − 1.91·97-s + 3.82·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.18322854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.18322854\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 239 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 1681 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T + 4050 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4561 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 80 T + 3200 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8126 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 150 T + 11250 T^{2} + 150 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12817 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 186 T + 17298 T^{2} + 186 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.064254668673264609512320466262, −9.017021032980945634453217683498, −8.673337176964867009928334614645, −8.466202815995631732741583056008, −7.999434943071348105132127645641, −7.25560286766133843043679606978, −6.96276543163024175912425706686, −6.46513023786755704404913687959, −6.19523306586739773063441009447, −5.95898469235243745750158087346, −5.09118311833027493930434550570, −4.96749689281710059397663867318, −4.36415098845163254722425504967, −4.15639351753944219496419102689, −3.33302332058225855903691482541, −3.11221640934703769996727182553, −1.84804883416187798434616150151, −1.76748373342262451484331766082, −1.15960075818472685750736142838, −1.13383421009418118627751563564,
1.13383421009418118627751563564, 1.15960075818472685750736142838, 1.76748373342262451484331766082, 1.84804883416187798434616150151, 3.11221640934703769996727182553, 3.33302332058225855903691482541, 4.15639351753944219496419102689, 4.36415098845163254722425504967, 4.96749689281710059397663867318, 5.09118311833027493930434550570, 5.95898469235243745750158087346, 6.19523306586739773063441009447, 6.46513023786755704404913687959, 6.96276543163024175912425706686, 7.25560286766133843043679606978, 7.999434943071348105132127645641, 8.466202815995631732741583056008, 8.673337176964867009928334614645, 9.017021032980945634453217683498, 9.064254668673264609512320466262
