L(s) = 1 | + 2-s + 0.879·3-s + 4-s + 0.532·5-s + 0.879·6-s + 7-s + 8-s − 2.22·9-s + 0.532·10-s − 2.87·11-s + 0.879·12-s − 0.347·13-s + 14-s + 0.467·15-s + 16-s − 4.59·17-s − 2.22·18-s + 0.532·20-s + 0.879·21-s − 2.87·22-s + 2.45·23-s + 0.879·24-s − 4.71·25-s − 0.347·26-s − 4.59·27-s + 28-s − 9.41·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.507·3-s + 0.5·4-s + 0.237·5-s + 0.359·6-s + 0.377·7-s + 0.353·8-s − 0.742·9-s + 0.168·10-s − 0.868·11-s + 0.253·12-s − 0.0963·13-s + 0.267·14-s + 0.120·15-s + 0.250·16-s − 1.11·17-s − 0.524·18-s + 0.118·20-s + 0.191·21-s − 0.613·22-s + 0.511·23-s + 0.179·24-s − 0.943·25-s − 0.0681·26-s − 0.884·27-s + 0.188·28-s − 1.74·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.879T + 3T^{2} \) |
| 5 | \( 1 - 0.532T + 5T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 0.347T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 + 9.06T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 9.92T + 73T^{2} \) |
| 79 | \( 1 + 0.263T + 79T^{2} \) |
| 83 | \( 1 - 3.95T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.76189957352163679646401629649, −7.24211030308274160518927988843, −6.31385513891749868783019038365, −5.47208861161116404306707135551, −5.12010697129637766479716975541, −4.03050461308935306128756949944, −3.35475878394219619750942694072, −2.37595654697367703214665232638, −1.88101220165856087921501759000, 0,
1.88101220165856087921501759000, 2.37595654697367703214665232638, 3.35475878394219619750942694072, 4.03050461308935306128756949944, 5.12010697129637766479716975541, 5.47208861161116404306707135551, 6.31385513891749868783019038365, 7.24211030308274160518927988843, 7.76189957352163679646401629649