| L(s) = 1 | − 2-s − 2.61·3-s + 4-s − 5-s + 2.61·6-s + 2.75·7-s − 8-s + 3.85·9-s + 10-s − 5.50·11-s − 2.61·12-s − 5.53·13-s − 2.75·14-s + 2.61·15-s + 16-s − 1.82·17-s − 3.85·18-s − 3.92·19-s − 20-s − 7.21·21-s + 5.50·22-s + 6.55·23-s + 2.61·24-s + 25-s + 5.53·26-s − 2.23·27-s + 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.51·3-s + 0.5·4-s − 0.447·5-s + 1.06·6-s + 1.04·7-s − 0.353·8-s + 1.28·9-s + 0.316·10-s − 1.65·11-s − 0.755·12-s − 1.53·13-s − 0.736·14-s + 0.675·15-s + 0.250·16-s − 0.443·17-s − 0.908·18-s − 0.899·19-s − 0.223·20-s − 1.57·21-s + 1.17·22-s + 1.36·23-s + 0.534·24-s + 0.200·25-s + 1.08·26-s − 0.430·27-s + 0.520·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 - 6.55T + 23T^{2} \) |
| 29 | \( 1 + 6.20T + 29T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 + 0.654T + 89T^{2} \) |
| 97 | \( 1 + 3.09T + 97T^{2} \) |
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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
−7.28328192695994518275405206954, −6.94170997499117608438663318677, −5.87217143299941812752276489776, −5.18300143772428945905137809244, −4.95037502641527639899289999546, −4.09899399613640469832903910292, −2.66991120431015712459335874288, −2.07351113283968610525022193160, −0.77252739109737169480148270696, 0,
0.77252739109737169480148270696, 2.07351113283968610525022193160, 2.66991120431015712459335874288, 4.09899399613640469832903910292, 4.95037502641527639899289999546, 5.18300143772428945905137809244, 5.87217143299941812752276489776, 6.94170997499117608438663318677, 7.28328192695994518275405206954
