L(s) = 1 | − 2.60·2-s − 3.03·3-s + 4.76·4-s + 7.87·6-s − 1.49·7-s − 7.18·8-s + 6.18·9-s − 4.55·11-s − 14.4·12-s − 6.26·13-s + 3.89·14-s + 9.15·16-s + 5.97·17-s − 16.0·18-s − 3.57·19-s + 4.54·21-s + 11.8·22-s − 6.76·23-s + 21.7·24-s + 16.2·26-s − 9.64·27-s − 7.13·28-s − 3.27·29-s − 4.86·31-s − 9.44·32-s + 13.7·33-s − 15.5·34-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 1.74·3-s + 2.38·4-s + 3.21·6-s − 0.566·7-s − 2.53·8-s + 2.06·9-s − 1.37·11-s − 4.16·12-s − 1.73·13-s + 1.04·14-s + 2.28·16-s + 1.44·17-s − 3.78·18-s − 0.821·19-s + 0.991·21-s + 2.52·22-s − 1.41·23-s + 4.44·24-s + 3.19·26-s − 1.85·27-s − 1.34·28-s − 0.608·29-s − 0.873·31-s − 1.66·32-s + 2.40·33-s − 2.66·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 6.26T + 13T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 - 9.31T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 + 8.42T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 8.77T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + 1.35T + 73T^{2} \) |
| 79 | \( 1 - 0.518T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 6.25T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−7.70906800271208865892762323990, −7.16910904277480280403858146421, −6.51570638921608941471122411039, −5.64840937177559347970550488758, −5.36487535564952569057754276612, −4.15114878315347296511912153062, −2.71811045843931575551802159210, −1.96356440451639001936518720872, −0.65448221006039776342440292006, 0,
0.65448221006039776342440292006, 1.96356440451639001936518720872, 2.71811045843931575551802159210, 4.15114878315347296511912153062, 5.36487535564952569057754276612, 5.64840937177559347970550488758, 6.51570638921608941471122411039, 7.16910904277480280403858146421, 7.70906800271208865892762323990