| L(s) = 1 | + 2.84·3-s + 3.71·5-s − 2.07·7-s + 5.07·9-s + 2.95·11-s + 0.984·13-s + 10.5·15-s − 17-s + 5.56·19-s − 5.88·21-s + 2.78·23-s + 8.80·25-s + 5.88·27-s − 5.67·29-s − 2.42·31-s + 8.38·33-s − 7.69·35-s + 3.60·37-s + 2.79·39-s − 9.74·41-s + 5.55·43-s + 18.8·45-s − 9.44·47-s − 2.70·49-s − 2.84·51-s + 6.31·53-s + 10.9·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s + 1.66·5-s − 0.782·7-s + 1.69·9-s + 0.890·11-s + 0.272·13-s + 2.72·15-s − 0.242·17-s + 1.27·19-s − 1.28·21-s + 0.581·23-s + 1.76·25-s + 1.13·27-s − 1.05·29-s − 0.435·31-s + 1.46·33-s − 1.30·35-s + 0.592·37-s + 0.447·39-s − 1.52·41-s + 0.847·43-s + 2.80·45-s − 1.37·47-s − 0.386·49-s − 0.397·51-s + 0.867·53-s + 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.711026330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.711026330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 - 0.984T + 13T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 9.39T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 0.0199T + 97T^{2} \) |
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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
−7.989242865117198142694636159100, −6.93929399038553566208171126535, −6.73579706857217662635878524151, −5.76127038202875882195219763688, −5.14008134668756701379160110265, −3.94416633659818366117738156713, −3.35370074237017851768719381848, −2.67707852354149128692523915582, −1.91162750438861092108108861793, −1.21370427427355314239694490088,
1.21370427427355314239694490088, 1.91162750438861092108108861793, 2.67707852354149128692523915582, 3.35370074237017851768719381848, 3.94416633659818366117738156713, 5.14008134668756701379160110265, 5.76127038202875882195219763688, 6.73579706857217662635878524151, 6.93929399038553566208171126535, 7.989242865117198142694636159100
