| L(s) = 1 | − 3·5-s − 7-s + 7·13-s − 2·17-s − 19-s + 6·23-s + 4·25-s + 7·29-s − 10·31-s + 3·35-s − 7·37-s + 6·41-s − 8·43-s − 13·47-s + 49-s − 10·53-s − 9·59-s − 6·61-s − 21·65-s + 67-s + 6·71-s + 13·73-s + 8·79-s + 18·83-s + 6·85-s + 2·89-s − 7·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.94·13-s − 0.485·17-s − 0.229·19-s + 1.25·23-s + 4/5·25-s + 1.29·29-s − 1.79·31-s + 0.507·35-s − 1.15·37-s + 0.937·41-s − 1.21·43-s − 1.89·47-s + 1/7·49-s − 1.37·53-s − 1.17·59-s − 0.768·61-s − 2.60·65-s + 0.122·67-s + 0.712·71-s + 1.52·73-s + 0.900·79-s + 1.97·83-s + 0.650·85-s + 0.211·89-s − 0.733·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196282772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196282772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.22298425415609, −13.81287303785597, −13.15292762535796, −12.82464897458573, −12.29216886351600, −11.71520560769137, −11.11396476941870, −10.89019942891149, −10.52412469324894, −9.443523297878543, −9.172993586874389, −8.466635286126352, −8.183633135466304, −7.633261304657173, −6.813230575298767, −6.584952458913486, −5.980328150056215, −5.066424182716045, −4.681030099821120, −3.883807118538459, −3.355099503715014, −3.211020151093821, −1.978016129180614, −1.247975197097404, −0.4023853670310819,
0.4023853670310819, 1.247975197097404, 1.978016129180614, 3.211020151093821, 3.355099503715014, 3.883807118538459, 4.681030099821120, 5.066424182716045, 5.980328150056215, 6.584952458913486, 6.813230575298767, 7.633261304657173, 8.183633135466304, 8.466635286126352, 9.172993586874389, 9.443523297878543, 10.52412469324894, 10.89019942891149, 11.11396476941870, 11.71520560769137, 12.29216886351600, 12.82464897458573, 13.15292762535796, 13.81287303785597, 14.22298425415609
