| L(s) = 1 | + 3·2-s + 3·4-s − 5-s − 3·8-s + 3·9-s − 3·10-s − 13·16-s + 9·18-s − 3·20-s + 6·23-s + 25-s + 11·31-s − 15·32-s + 9·36-s + 6·37-s + 3·40-s + 6·41-s − 6·43-s − 3·45-s + 18·46-s − 5·49-s + 3·50-s − 9·59-s + 22·61-s + 33·62-s + 3·64-s − 9·72-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 0.447·5-s − 1.06·8-s + 9-s − 0.948·10-s − 3.25·16-s + 2.12·18-s − 0.670·20-s + 1.25·23-s + 1/5·25-s + 1.97·31-s − 2.65·32-s + 3/2·36-s + 0.986·37-s + 0.474·40-s + 0.937·41-s − 0.914·43-s − 0.447·45-s + 2.65·46-s − 5/7·49-s + 0.424·50-s − 1.17·59-s + 2.81·61-s + 4.19·62-s + 3/8·64-s − 1.06·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.004132916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.004132916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 + T \) |
| 41 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
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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
−9.064848201710220808958999497478, −8.560929622154225128644072787096, −8.044097957776542263614043625340, −7.48074167359819629414604573722, −6.87291152247822502332852384645, −6.35306506321535767527110085903, −6.16855855214014351674486151297, −5.18026335793406642179265983708, −5.05180578684913252373118318964, −4.50177544140202790221628980597, −4.09512507221147170994262635350, −3.57601482203249611776571811659, −2.98797403882698064697861969703, −2.38895089274862251525364903201, −0.957906986399994902061369365314,
0.957906986399994902061369365314, 2.38895089274862251525364903201, 2.98797403882698064697861969703, 3.57601482203249611776571811659, 4.09512507221147170994262635350, 4.50177544140202790221628980597, 5.05180578684913252373118318964, 5.18026335793406642179265983708, 6.16855855214014351674486151297, 6.35306506321535767527110085903, 6.87291152247822502332852384645, 7.48074167359819629414604573722, 8.044097957776542263614043625340, 8.560929622154225128644072787096, 9.064848201710220808958999497478
