L(s) = 1 | + 2.17·2-s − 0.456·3-s + 2.73·4-s − 0.352·5-s − 0.992·6-s − 3.48·7-s + 1.59·8-s − 2.79·9-s − 0.767·10-s − 1.91·11-s − 1.24·12-s + 13-s − 7.57·14-s + 0.160·15-s − 1.99·16-s + 0.154·17-s − 6.07·18-s − 0.270·19-s − 0.964·20-s + 1.58·21-s − 4.17·22-s + 4.69·23-s − 0.729·24-s − 4.87·25-s + 2.17·26-s + 2.64·27-s − 9.51·28-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 0.263·3-s + 1.36·4-s − 0.157·5-s − 0.405·6-s − 1.31·7-s + 0.564·8-s − 0.930·9-s − 0.242·10-s − 0.578·11-s − 0.360·12-s + 0.277·13-s − 2.02·14-s + 0.0415·15-s − 0.497·16-s + 0.0374·17-s − 1.43·18-s − 0.0621·19-s − 0.215·20-s + 0.346·21-s − 0.889·22-s + 0.978·23-s − 0.148·24-s − 0.975·25-s + 0.426·26-s + 0.508·27-s − 1.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 0.456T + 3T^{2} \) |
| 5 | \( 1 + 0.352T + 5T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 17 | \( 1 - 0.154T + 17T^{2} \) |
| 19 | \( 1 + 0.270T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 + 0.575T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 - 5.16T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.09T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.209534007776034967112641854633, −8.413183761764842380708072982811, −7.07099027782026518558125585940, −6.55856531493107258290775751638, −5.53261487706670029085075266997, −5.27094094229904603202719386820, −3.83485764490517842438069656103, −3.31738719783720970882584754772, −2.37234865057065501026195491072, 0,
2.37234865057065501026195491072, 3.31738719783720970882584754772, 3.83485764490517842438069656103, 5.27094094229904603202719386820, 5.53261487706670029085075266997, 6.55856531493107258290775751638, 7.07099027782026518558125585940, 8.413183761764842380708072982811, 9.209534007776034967112641854633