| L(s) = 1 | − 49.1·3-s + 1.43e3·7-s − 1.72e4·9-s + 8.37e4·11-s − 1.45e5·13-s + 3.07e3·17-s + 6.53e5·19-s − 7.06e4·21-s − 1.89e6·23-s + 1.81e6·27-s + 4.12e6·29-s − 5.26e6·31-s − 4.11e6·33-s + 9.17e6·37-s + 7.16e6·39-s − 8.05e6·41-s − 4.82e5·43-s − 6.00e6·47-s − 3.82e7·49-s − 1.51e5·51-s + 8.82e7·53-s − 3.21e7·57-s − 9.30e7·59-s + 1.00e8·61-s − 2.48e7·63-s + 8.02e7·67-s + 9.31e7·69-s + ⋯ |
| L(s) = 1 | − 0.350·3-s + 0.226·7-s − 0.877·9-s + 1.72·11-s − 1.41·13-s + 0.00894·17-s + 1.14·19-s − 0.0792·21-s − 1.41·23-s + 0.658·27-s + 1.08·29-s − 1.02·31-s − 0.604·33-s + 0.804·37-s + 0.496·39-s − 0.445·41-s − 0.0215·43-s − 0.179·47-s − 0.948·49-s − 0.00313·51-s + 1.53·53-s − 0.403·57-s − 1.00·59-s + 0.932·61-s − 0.198·63-s + 0.486·67-s + 0.494·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 49.1T + 1.96e4T^{2} \) |
| 7 | \( 1 - 1.43e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.37e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.45e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.07e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.53e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.89e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.26e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.17e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 8.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.82e5T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.00e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.82e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.02e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.01e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.30e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.394695459249064703994680314255, −8.431754402398703679409959261638, −7.39116268123370677447574468585, −6.45603304192649450628452000940, −5.54429736315856880081115404828, −4.56446756052100514932811190712, −3.46772009258047458105297905527, −2.28233157411255653959127308039, −1.10468573283294469875007320341, 0,
1.10468573283294469875007320341, 2.28233157411255653959127308039, 3.46772009258047458105297905527, 4.56446756052100514932811190712, 5.54429736315856880081115404828, 6.45603304192649450628452000940, 7.39116268123370677447574468585, 8.431754402398703679409959261638, 9.394695459249064703994680314255
