L(s) = 1 | + 1.89·2-s + 3-s + 1.60·4-s − 0.521·5-s + 1.89·6-s + 0.438·7-s − 0.753·8-s + 9-s − 0.989·10-s − 3.48·11-s + 1.60·12-s − 13-s + 0.832·14-s − 0.521·15-s − 4.63·16-s − 2.03·17-s + 1.89·18-s − 2.05·19-s − 0.835·20-s + 0.438·21-s − 6.61·22-s + 0.601·23-s − 0.753·24-s − 4.72·25-s − 1.89·26-s + 27-s + 0.703·28-s + ⋯ |
L(s) = 1 | + 1.34·2-s + 0.577·3-s + 0.801·4-s − 0.233·5-s + 0.774·6-s + 0.165·7-s − 0.266·8-s + 0.333·9-s − 0.312·10-s − 1.05·11-s + 0.462·12-s − 0.277·13-s + 0.222·14-s − 0.134·15-s − 1.15·16-s − 0.493·17-s + 0.447·18-s − 0.471·19-s − 0.186·20-s + 0.0957·21-s − 1.41·22-s + 0.125·23-s − 0.153·24-s − 0.945·25-s − 0.372·26-s + 0.192·27-s + 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 5 | \( 1 + 0.521T + 5T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 - 0.601T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 + 0.0798T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 0.354T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 - 3.57T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 9.54T + 71T^{2} \) |
| 73 | \( 1 + 7.45T + 73T^{2} \) |
| 79 | \( 1 - 1.15T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.966908153576950431919298541893, −7.28378280168261764726640630756, −6.50486142137363726863707915742, −5.53795490803306071049413296167, −5.08571956354464189712233573358, −4.12847383543964962370219108455, −3.64532710993139985284266155171, −2.64039726816284664509449366410, −2.00847986827187506144225038297, 0,
2.00847986827187506144225038297, 2.64039726816284664509449366410, 3.64532710993139985284266155171, 4.12847383543964962370219108455, 5.08571956354464189712233573358, 5.53795490803306071049413296167, 6.50486142137363726863707915742, 7.28378280168261764726640630756, 7.966908153576950431919298541893