L(s) = 1 | + 1.77·2-s + 0.796·3-s + 1.14·4-s + 1.41·6-s + 3.30·7-s − 1.52·8-s − 2.36·9-s + 2.56·11-s + 0.909·12-s − 4.28·13-s + 5.86·14-s − 4.97·16-s − 3.96·17-s − 4.19·18-s − 3.96·19-s + 2.63·21-s + 4.54·22-s − 2.74·23-s − 1.21·24-s − 7.59·26-s − 4.27·27-s + 3.77·28-s − 3.80·29-s + 4.48·31-s − 5.78·32-s + 2.04·33-s − 7.02·34-s + ⋯ |
L(s) = 1 | + 1.25·2-s + 0.459·3-s + 0.571·4-s + 0.576·6-s + 1.24·7-s − 0.537·8-s − 0.788·9-s + 0.773·11-s + 0.262·12-s − 1.18·13-s + 1.56·14-s − 1.24·16-s − 0.961·17-s − 0.988·18-s − 0.908·19-s + 0.574·21-s + 0.969·22-s − 0.573·23-s − 0.247·24-s − 1.48·26-s − 0.822·27-s + 0.713·28-s − 0.707·29-s + 0.805·31-s − 1.02·32-s + 0.355·33-s − 1.20·34-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 | \( 1 - 1.77T + 2T^{2} \) |
| 3 | \( 1 - 0.796T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 - 4.48T + 31T^{2} \) |
| 37 | \( 1 - 1.87T + 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 - 0.780T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 9.66T + 97T^{2} \) |
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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
−7.80137449514622735118660390945, −6.77803343635263888717874298029, −6.19576799153007094414909584715, −5.36007077908596712862372552135, −4.67528107749447920908103020959, −4.26208892120847844577305194727, −3.35094959654348727974554864814, −2.44266361713905227735594007456, −1.85160971037016328185443785296, 0,
1.85160971037016328185443785296, 2.44266361713905227735594007456, 3.35094959654348727974554864814, 4.26208892120847844577305194727, 4.67528107749447920908103020959, 5.36007077908596712862372552135, 6.19576799153007094414909584715, 6.77803343635263888717874298029, 7.80137449514622735118660390945