L(s) = 1 | + 2.00·2-s + 1.95·3-s + 2.02·4-s + 5-s + 3.92·6-s + 0.0561·8-s + 0.816·9-s + 2.00·10-s − 3.58·11-s + 3.96·12-s − 6.74·13-s + 1.95·15-s − 3.94·16-s − 1.85·17-s + 1.63·18-s + 4.97·19-s + 2.02·20-s − 7.20·22-s − 7.86·23-s + 0.109·24-s + 25-s − 13.5·26-s − 4.26·27-s − 3.79·29-s + 3.92·30-s − 31-s − 8.02·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.12·3-s + 1.01·4-s + 0.447·5-s + 1.60·6-s + 0.0198·8-s + 0.272·9-s + 0.634·10-s − 1.08·11-s + 1.14·12-s − 1.87·13-s + 0.504·15-s − 0.985·16-s − 0.449·17-s + 0.386·18-s + 1.14·19-s + 0.453·20-s − 1.53·22-s − 1.64·23-s + 0.0224·24-s + 0.200·25-s − 2.65·26-s − 0.820·27-s − 0.705·29-s + 0.715·30-s − 0.179·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 3 | \( 1 - 1.95T + 3T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 + 7.86T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 37 | \( 1 + 1.00T + 37T^{2} \) |
| 41 | \( 1 - 6.71T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.74T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.57T + 83T^{2} \) |
| 89 | \( 1 - 0.451T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.58971399903889724546781461533, −6.82139684495464113580006270063, −5.72832585111596420431098151426, −5.45285080026715950055709787577, −4.62789255136857909541150473334, −3.94155684220317159678596958198, −3.06055196998190420304486392396, −2.48406689340560065371667837835, −2.07613260840442186571114188026, 0,
2.07613260840442186571114188026, 2.48406689340560065371667837835, 3.06055196998190420304486392396, 3.94155684220317159678596958198, 4.62789255136857909541150473334, 5.45285080026715950055709787577, 5.72832585111596420431098151426, 6.82139684495464113580006270063, 7.58971399903889724546781461533