| L(s) = 1 | − 2.27·2-s + 3.19·4-s + 1.72·5-s − 5.05·7-s − 2.71·8-s − 3.94·10-s − 11-s − 6.17·13-s + 11.5·14-s − 0.188·16-s + 2.57·17-s − 5.04·19-s + 5.52·20-s + 2.27·22-s − 4.47·23-s − 2.00·25-s + 14.0·26-s − 16.1·28-s − 4.65·29-s − 3.09·31-s + 5.86·32-s − 5.87·34-s − 8.74·35-s + 10.7·37-s + 11.4·38-s − 4.70·40-s − 6.60·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.59·4-s + 0.773·5-s − 1.91·7-s − 0.961·8-s − 1.24·10-s − 0.301·11-s − 1.71·13-s + 3.07·14-s − 0.0471·16-s + 0.625·17-s − 1.15·19-s + 1.23·20-s + 0.485·22-s − 0.933·23-s − 0.401·25-s + 2.75·26-s − 3.05·28-s − 0.863·29-s − 0.556·31-s + 1.03·32-s − 1.00·34-s − 1.47·35-s + 1.77·37-s + 1.86·38-s − 0.743·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1160845019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1160845019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 13 | \( 1 + 6.17T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 6.60T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 + 6.55T + 59T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 1.35T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.057934095087446044140678136923, −7.52098708805856482196257747575, −6.77366203295079599083956279017, −6.25029047152940290689872838331, −5.57414770160559395284925203107, −4.40720600524167497309087516972, −3.24095579194273380364877213514, −2.45569966971129493840102135820, −1.79862521854351385485678057013, −0.21262736666465604858283523229,
0.21262736666465604858283523229, 1.79862521854351385485678057013, 2.45569966971129493840102135820, 3.24095579194273380364877213514, 4.40720600524167497309087516972, 5.57414770160559395284925203107, 6.25029047152940290689872838331, 6.77366203295079599083956279017, 7.52098708805856482196257747575, 8.057934095087446044140678136923
