L(s) = 1 | − 1.59·2-s + 0.0342·3-s + 0.533·4-s − 0.0545·6-s − 1.01·7-s + 2.33·8-s − 2.99·9-s + 5.74·11-s + 0.0182·12-s + 4.77·13-s + 1.62·14-s − 4.78·16-s − 2.20·17-s + 4.77·18-s + 7.81·19-s − 0.0348·21-s − 9.13·22-s − 3.98·23-s + 0.0800·24-s − 7.60·26-s − 0.205·27-s − 0.543·28-s + 0.409·29-s − 9.04·31-s + 2.94·32-s + 0.196·33-s + 3.51·34-s + ⋯ |
L(s) = 1 | − 1.12·2-s + 0.0197·3-s + 0.266·4-s − 0.0222·6-s − 0.384·7-s + 0.825·8-s − 0.999·9-s + 1.73·11-s + 0.00528·12-s + 1.32·13-s + 0.433·14-s − 1.19·16-s − 0.535·17-s + 1.12·18-s + 1.79·19-s − 0.00761·21-s − 1.94·22-s − 0.830·23-s + 0.0163·24-s − 1.49·26-s − 0.0395·27-s − 0.102·28-s + 0.0760·29-s − 1.62·31-s + 0.520·32-s + 0.0342·33-s + 0.602·34-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 | \( 1 + 1.59T + 2T^{2} \) |
| 3 | \( 1 - 0.0342T + 3T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 + 2.20T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 0.409T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 + 7.42T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 0.457T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 2.52T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.967351009617663533742187963034, −7.03816819971506628785986177850, −6.53986867071436209349816956924, −5.73136048885822582825330007207, −4.91258784221702439519822569372, −3.62826003286511059392827493033, −3.50032429284690900962620411537, −1.88595887928303975432192381460, −1.20275576882208343181983826210, 0,
1.20275576882208343181983826210, 1.88595887928303975432192381460, 3.50032429284690900962620411537, 3.62826003286511059392827493033, 4.91258784221702439519822569372, 5.73136048885822582825330007207, 6.53986867071436209349816956924, 7.03816819971506628785986177850, 7.967351009617663533742187963034