L(s) = 1 | + 0.201·2-s − 0.798·3-s − 1.95·4-s − 0.160·6-s + 1.75·7-s − 0.798·8-s − 2.36·9-s + 1.56·12-s + 2.79·13-s + 0.354·14-s + 3.75·16-s − 5.71·17-s − 0.476·18-s + 0.757·19-s − 1.40·21-s + 3.71·23-s + 0.637·24-s + 0.564·26-s + 4.28·27-s − 3.44·28-s − 6.95·29-s + 7.51·31-s + 2.35·32-s − 1.15·34-s + 4.62·36-s + 1.79·37-s + 0.152·38-s + ⋯ |
L(s) = 1 | + 0.142·2-s − 0.460·3-s − 0.979·4-s − 0.0657·6-s + 0.664·7-s − 0.282·8-s − 0.787·9-s + 0.451·12-s + 0.776·13-s + 0.0947·14-s + 0.939·16-s − 1.38·17-s − 0.112·18-s + 0.173·19-s − 0.306·21-s + 0.775·23-s + 0.130·24-s + 0.110·26-s + 0.823·27-s − 0.650·28-s − 1.29·29-s + 1.34·31-s + 0.416·32-s − 0.197·34-s + 0.771·36-s + 0.295·37-s + 0.0247·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.201T + 2T^{2} \) |
| 3 | \( 1 + 0.798T + 3T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 - 0.757T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 + 6.95T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 4.51T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 + 0.362T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.433008641309797646801855951346, −7.81402636825235085385113696342, −6.63257817987787908708228056085, −5.97183473333665610529601423636, −5.14312808268528721218147677148, −4.60522498063897041483420894302, −3.72158532427529379080509819658, −2.68311638200442585714002187331, −1.28080934590070722849519201952, 0,
1.28080934590070722849519201952, 2.68311638200442585714002187331, 3.72158532427529379080509819658, 4.60522498063897041483420894302, 5.14312808268528721218147677148, 5.97183473333665610529601423636, 6.63257817987787908708228056085, 7.81402636825235085385113696342, 8.433008641309797646801855951346