| L(s) = 1 | − 1.60·2-s + 0.565·4-s − 0.438·5-s + 0.237·7-s + 2.29·8-s + 0.702·10-s + 3.06·11-s + 3.96·13-s − 0.380·14-s − 4.81·16-s + 3.75·17-s − 19-s − 0.248·20-s − 4.91·22-s + 8.99·23-s − 4.80·25-s − 6.35·26-s + 0.134·28-s − 6.02·29-s − 2.02·31-s + 3.11·32-s − 6.01·34-s − 0.104·35-s − 2.31·37-s + 1.60·38-s − 1.00·40-s − 2.82·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.282·4-s − 0.196·5-s + 0.0898·7-s + 0.812·8-s + 0.222·10-s + 0.925·11-s + 1.10·13-s − 0.101·14-s − 1.20·16-s + 0.910·17-s − 0.229·19-s − 0.0555·20-s − 1.04·22-s + 1.87·23-s − 0.961·25-s − 1.24·26-s + 0.0254·28-s − 1.11·29-s − 0.363·31-s + 0.550·32-s − 1.03·34-s − 0.0176·35-s − 0.380·37-s + 0.259·38-s − 0.159·40-s − 0.441·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 + 0.438T + 5T^{2} \) |
| 7 | \( 1 - 0.237T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 - 3.96T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 23 | \( 1 - 8.99T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 3.91T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
| 97 | \( 1 - 1.28T + 97T^{2} \) |
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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
−7.62325621330182908497623054349, −7.01085446106992927051177319032, −6.31583667354572527271672476786, −5.44340981752469408900581223348, −4.64597181125882685762111431603, −3.77874387020157238707075828905, −3.17834955706700421422504326375, −1.66302904213419373294642415875, −1.28128910307599791596158553282, 0,
1.28128910307599791596158553282, 1.66302904213419373294642415875, 3.17834955706700421422504326375, 3.77874387020157238707075828905, 4.64597181125882685762111431603, 5.44340981752469408900581223348, 6.31583667354572527271672476786, 7.01085446106992927051177319032, 7.62325621330182908497623054349
