L(s) = 1 | − 0.257·2-s − 3.28·3-s − 1.93·4-s + 0.843·6-s + 3.60·7-s + 1.01·8-s + 7.77·9-s − 1.43·11-s + 6.34·12-s + 2.88·13-s − 0.926·14-s + 3.60·16-s + 0.875·17-s − 1.99·18-s + 2.64·19-s − 11.8·21-s + 0.369·22-s − 4.70·23-s − 3.31·24-s − 0.741·26-s − 15.6·27-s − 6.97·28-s − 3.36·29-s − 0.807·31-s − 2.95·32-s + 4.72·33-s − 0.225·34-s + ⋯ |
L(s) = 1 | − 0.181·2-s − 1.89·3-s − 0.966·4-s + 0.344·6-s + 1.36·7-s + 0.357·8-s + 2.59·9-s − 0.433·11-s + 1.83·12-s + 0.799·13-s − 0.247·14-s + 0.901·16-s + 0.212·17-s − 0.470·18-s + 0.606·19-s − 2.58·21-s + 0.0788·22-s − 0.980·23-s − 0.677·24-s − 0.145·26-s − 3.01·27-s − 1.31·28-s − 0.624·29-s − 0.145·31-s − 0.521·32-s + 0.822·33-s − 0.0386·34-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 | \( 1 + 0.257T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 0.875T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 3.36T + 29T^{2} \) |
| 31 | \( 1 + 0.807T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 + 9.22T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 - 1.63T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 0.971T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
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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
−7.72984559915791712710207260836, −7.02942160324812105673729057581, −5.95280024907213385701753237690, −5.56545161829091838590410677281, −4.92036962147061894253252133937, −4.40550467822431416659296730629, −3.62366356208751776136915084261, −1.75365513431516678249786302452, −1.08681445983356670860840198757, 0,
1.08681445983356670860840198757, 1.75365513431516678249786302452, 3.62366356208751776136915084261, 4.40550467822431416659296730629, 4.92036962147061894253252133937, 5.56545161829091838590410677281, 5.95280024907213385701753237690, 7.02942160324812105673729057581, 7.72984559915791712710207260836