| L(s) = 1 | − 2.47·2-s + 4.13·4-s + 0.821·5-s − 3.73·7-s − 5.28·8-s − 2.03·10-s + 11-s + 6.75·13-s + 9.25·14-s + 4.82·16-s + 4.83·17-s − 7.06·19-s + 3.39·20-s − 2.47·22-s − 2.72·23-s − 4.32·25-s − 16.7·26-s − 15.4·28-s + 3.71·29-s + 3.15·31-s − 1.38·32-s − 11.9·34-s − 3.07·35-s − 3.39·37-s + 17.4·38-s − 4.34·40-s + 5.93·41-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.06·4-s + 0.367·5-s − 1.41·7-s − 1.86·8-s − 0.643·10-s + 0.301·11-s + 1.87·13-s + 2.47·14-s + 1.20·16-s + 1.17·17-s − 1.62·19-s + 0.759·20-s − 0.528·22-s − 0.568·23-s − 0.864·25-s − 3.28·26-s − 2.91·28-s + 0.690·29-s + 0.566·31-s − 0.244·32-s − 2.05·34-s − 0.518·35-s − 0.557·37-s + 2.83·38-s − 0.687·40-s + 0.927·41-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
| \(L(1)\) |
\(\approx\) |
\(0.7307040913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7307040913\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 - 0.821T + 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 7.67T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 + 6.87T + 73T^{2} \) |
| 79 | \( 1 + 0.687T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.192551014556632534754394236865, −7.63239006602849926737609123087, −6.59648364265061884649392594883, −6.25014816230306588764019783360, −5.82889456403688200905610327203, −4.15294853448158026327305472888, −3.40947742499536528039922375790, −2.48742445085508728961152224658, −1.52223020107757589899648869269, −0.60940692079471002219076210615,
0.60940692079471002219076210615, 1.52223020107757589899648869269, 2.48742445085508728961152224658, 3.40947742499536528039922375790, 4.15294853448158026327305472888, 5.82889456403688200905610327203, 6.25014816230306588764019783360, 6.59648364265061884649392594883, 7.63239006602849926737609123087, 8.192551014556632534754394236865
