L(s) = 1 | + 1.70·2-s + 1.38·3-s + 0.922·4-s + 2.36·6-s + 5.05·7-s − 1.84·8-s − 1.08·9-s − 5.25·11-s + 1.27·12-s − 2.84·13-s + 8.64·14-s − 4.99·16-s − 6.16·17-s − 1.85·18-s + 1.47·19-s + 6.99·21-s − 8.97·22-s + 2.54·23-s − 2.54·24-s − 4.86·26-s − 5.65·27-s + 4.66·28-s − 9.53·29-s − 7.94·31-s − 4.85·32-s − 7.26·33-s − 10.5·34-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 0.798·3-s + 0.461·4-s + 0.965·6-s + 1.91·7-s − 0.651·8-s − 0.362·9-s − 1.58·11-s + 0.368·12-s − 0.788·13-s + 2.31·14-s − 1.24·16-s − 1.49·17-s − 0.437·18-s + 0.337·19-s + 1.52·21-s − 1.91·22-s + 0.530·23-s − 0.520·24-s − 0.953·26-s − 1.08·27-s + 0.882·28-s − 1.77·29-s − 1.42·31-s − 0.858·32-s − 1.26·33-s − 1.80·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 3 | \( 1 - 1.38T + 3T^{2} \) |
| 7 | \( 1 - 5.05T + 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 + 9.53T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 0.0179T + 43T^{2} \) |
| 47 | \( 1 + 5.95T + 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 + 3.28T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 9.27T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.70930895976585524473438883327, −7.22132089399516062175233504330, −5.91631377743820744190194327971, −5.28855254936767950508619439922, −4.85562574779620629556051030956, −4.21678893238888201551253700625, −3.24771097223650930942178843060, −2.37934315239698284470717727171, −2.00521797972959947293540476323, 0,
2.00521797972959947293540476323, 2.37934315239698284470717727171, 3.24771097223650930942178843060, 4.21678893238888201551253700625, 4.85562574779620629556051030956, 5.28855254936767950508619439922, 5.91631377743820744190194327971, 7.22132089399516062175233504330, 7.70930895976585524473438883327