| L(s) = 1 | − 0.964·2-s − 1.07·4-s + 0.686·5-s − 3.77·7-s + 2.96·8-s − 0.662·10-s − 2.33·11-s − 5.60·13-s + 3.63·14-s − 0.713·16-s + 0.836·17-s + 3.09·19-s − 0.735·20-s + 2.24·22-s + 7.69·23-s − 4.52·25-s + 5.40·26-s + 4.03·28-s + 8.52·29-s + 7.78·31-s − 5.23·32-s − 0.806·34-s − 2.59·35-s + 8.23·37-s − 2.98·38-s + 2.03·40-s + 8.33·41-s + ⋯ |
| L(s) = 1 | − 0.681·2-s − 0.535·4-s + 0.307·5-s − 1.42·7-s + 1.04·8-s − 0.209·10-s − 0.703·11-s − 1.55·13-s + 0.972·14-s − 0.178·16-s + 0.202·17-s + 0.709·19-s − 0.164·20-s + 0.479·22-s + 1.60·23-s − 0.905·25-s + 1.06·26-s + 0.763·28-s + 1.58·29-s + 1.39·31-s − 0.925·32-s − 0.138·34-s − 0.438·35-s + 1.35·37-s − 0.483·38-s + 0.321·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 0.964T + 2T^{2} \) |
| 5 | \( 1 - 0.686T + 5T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 - 0.836T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 - 8.52T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 8.23T + 37T^{2} \) |
| 41 | \( 1 - 8.33T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 6.81T + 67T^{2} \) |
| 71 | \( 1 + 3.58T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 3.01T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 0.288T + 97T^{2} \) |
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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
−7.937093737416101197373780241244, −7.61039351728522341635481912701, −6.69570435600953860988807495449, −5.95798754513204210140233947380, −4.94403494141668146071083520804, −4.48634884032509304826297475629, −3.08220441984431359329066697771, −2.65159404356373023952282380765, −1.05477339267568639778289030467, 0,
1.05477339267568639778289030467, 2.65159404356373023952282380765, 3.08220441984431359329066697771, 4.48634884032509304826297475629, 4.94403494141668146071083520804, 5.95798754513204210140233947380, 6.69570435600953860988807495449, 7.61039351728522341635481912701, 7.937093737416101197373780241244
