L(s) = 1 | − 1.98·2-s + 2.93·3-s + 1.94·4-s − 5.82·6-s − 3.87·7-s + 0.108·8-s + 5.60·9-s + 5.43·11-s + 5.70·12-s + 4.51·13-s + 7.70·14-s − 4.10·16-s − 3.56·17-s − 11.1·18-s − 4.95·19-s − 11.3·21-s − 10.7·22-s − 8.91·23-s + 0.319·24-s − 8.96·26-s + 7.64·27-s − 7.54·28-s + 2.56·29-s − 1.87·31-s + 7.93·32-s + 15.9·33-s + 7.08·34-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 1.69·3-s + 0.972·4-s − 2.37·6-s − 1.46·7-s + 0.0385·8-s + 1.86·9-s + 1.63·11-s + 1.64·12-s + 1.25·13-s + 2.05·14-s − 1.02·16-s − 0.864·17-s − 2.62·18-s − 1.13·19-s − 2.48·21-s − 2.30·22-s − 1.85·23-s + 0.0652·24-s − 1.75·26-s + 1.47·27-s − 1.42·28-s + 0.475·29-s − 0.336·31-s + 1.40·32-s + 2.77·33-s + 1.21·34-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 | \( 1 + 1.98T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 + 8.91T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 8.26T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 0.538T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 8.98T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−8.139857121535444819743625379883, −7.17887683802580634514950587224, −6.54724067055149404929587488534, −6.21821269002215278370602250604, −4.25370842288032477481475297349, −3.87333571549238204721776260089, −3.12542083016795341578275885094, −2.03833055038624911345205946778, −1.49024352183833019477856745664, 0,
1.49024352183833019477856745664, 2.03833055038624911345205946778, 3.12542083016795341578275885094, 3.87333571549238204721776260089, 4.25370842288032477481475297349, 6.21821269002215278370602250604, 6.54724067055149404929587488534, 7.17887683802580634514950587224, 8.139857121535444819743625379883