| L(s) = 1 | − 2.07·2-s + 2.28·4-s + 2.79·5-s − 4.09·7-s − 0.590·8-s − 5.78·10-s + 3.90·11-s + 6.98·13-s + 8.47·14-s − 3.34·16-s + 1.86·17-s + 19-s + 6.39·20-s − 8.07·22-s − 2.28·23-s + 2.82·25-s − 14.4·26-s − 9.35·28-s − 9.96·29-s + 9.53·31-s + 8.11·32-s − 3.85·34-s − 11.4·35-s + 0.376·37-s − 2.07·38-s − 1.65·40-s − 11.1·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.14·4-s + 1.25·5-s − 1.54·7-s − 0.208·8-s − 1.83·10-s + 1.17·11-s + 1.93·13-s + 2.26·14-s − 0.837·16-s + 0.452·17-s + 0.229·19-s + 1.42·20-s − 1.72·22-s − 0.476·23-s + 0.564·25-s − 2.83·26-s − 1.76·28-s − 1.85·29-s + 1.71·31-s + 1.43·32-s − 0.661·34-s − 1.93·35-s + 0.0619·37-s − 0.335·38-s − 0.260·40-s − 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242762743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242762743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 6.98T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 - 0.376T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 9.17T + 67T^{2} \) |
| 71 | \( 1 + 1.44T + 71T^{2} \) |
| 73 | \( 1 + 8.15T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 + 2.29T + 97T^{2} \) |
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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
−8.148820236257522281711898154236, −6.97418377345530038442572960683, −6.58637337499235722431554458934, −6.08631473654361401466564831133, −5.43496907981346122486540534658, −3.93524322370047905431093175956, −3.46676852093629195281390017331, −2.30283853437601299096097974734, −1.48544224212123603465508036421, −0.75006566058058918716665352182,
0.75006566058058918716665352182, 1.48544224212123603465508036421, 2.30283853437601299096097974734, 3.46676852093629195281390017331, 3.93524322370047905431093175956, 5.43496907981346122486540534658, 6.08631473654361401466564831133, 6.58637337499235722431554458934, 6.97418377345530038442572960683, 8.148820236257522281711898154236
