| L(s) = 1 | − 2·5-s − 4·7-s + 8·11-s + 8·17-s + 12·23-s + 2·25-s − 10·29-s + 12·31-s + 8·35-s − 6·37-s + 2·41-s − 8·47-s + 8·49-s − 16·55-s − 14·61-s + 8·67-s + 4·71-s + 22·73-s − 32·77-s + 20·79-s − 16·85-s + 12·89-s + 2·97-s − 20·101-s − 16·103-s + 8·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 2.41·11-s + 1.94·17-s + 2.50·23-s + 2/5·25-s − 1.85·29-s + 2.15·31-s + 1.35·35-s − 0.986·37-s + 0.312·41-s − 1.16·47-s + 8/7·49-s − 2.15·55-s − 1.79·61-s + 0.977·67-s + 0.474·71-s + 2.57·73-s − 3.64·77-s + 2.25·79-s − 1.73·85-s + 1.27·89-s + 0.203·97-s − 1.99·101-s − 1.57·103-s + 0.773·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642478717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642478717\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.73832354680183839005206663074, −10.57666994473032512311495035219, −9.712943332079620702904224880717, −9.399053246256000894384537368025, −9.374925459912937453641427683254, −8.854669748143288446263038339853, −8.091462973279646554223775897170, −7.86357144834143598832965002568, −7.17029040357322757241518491399, −6.72803835872275106961877198295, −6.53430347233623719086091638147, −6.09576564322952022389165580841, −5.21231917020507877968658723982, −4.96466095759736728144939614278, −3.90805101118437922388867547692, −3.75370463950568375824643881756, −3.30778476311261671731741503051, −2.78581035484535109265896794328, −1.42468603163083608634659134336, −0.814687359582873198965572374691,
0.814687359582873198965572374691, 1.42468603163083608634659134336, 2.78581035484535109265896794328, 3.30778476311261671731741503051, 3.75370463950568375824643881756, 3.90805101118437922388867547692, 4.96466095759736728144939614278, 5.21231917020507877968658723982, 6.09576564322952022389165580841, 6.53430347233623719086091638147, 6.72803835872275106961877198295, 7.17029040357322757241518491399, 7.86357144834143598832965002568, 8.091462973279646554223775897170, 8.854669748143288446263038339853, 9.374925459912937453641427683254, 9.399053246256000894384537368025, 9.712943332079620702904224880717, 10.57666994473032512311495035219, 10.73832354680183839005206663074
