| L(s) = 1 | − 2·4-s − 5-s + 7-s − 11-s − 4·13-s + 4·16-s − 6·17-s + 2·20-s − 4·25-s − 2·28-s − 4·29-s − 5·31-s − 35-s + 6·37-s − 7·41-s + 12·43-s + 2·44-s + 47-s + 49-s + 8·52-s − 53-s + 55-s − 2·59-s − 3·61-s − 8·64-s + 4·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s − 0.301·11-s − 1.10·13-s + 16-s − 1.45·17-s + 0.447·20-s − 4/5·25-s − 0.377·28-s − 0.742·29-s − 0.898·31-s − 0.169·35-s + 0.986·37-s − 1.09·41-s + 1.82·43-s + 0.301·44-s + 0.145·47-s + 1/7·49-s + 1.10·52-s − 0.137·53-s + 0.134·55-s − 0.260·59-s − 0.384·61-s − 64-s + 0.496·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.22767534055097, −12.94978800431873, −12.36706825344029, −12.06889972679673, −11.42573257944420, −10.97727205620922, −10.64087285629421, −9.957403790015003, −9.507208957206205, −9.163042147103560, −8.728911944873959, −8.070294067458238, −7.793641708566502, −7.300084927404473, −6.825429208442436, −6.042761482087240, −5.581206972236478, −5.098661175635597, −4.513874722376392, −4.237505323645477, −3.737583616320987, −3.014041742470457, −2.351271624651685, −1.868479179190921, −0.9966694882659946, 0, 0,
0.9966694882659946, 1.868479179190921, 2.351271624651685, 3.014041742470457, 3.737583616320987, 4.237505323645477, 4.513874722376392, 5.098661175635597, 5.581206972236478, 6.042761482087240, 6.825429208442436, 7.300084927404473, 7.793641708566502, 8.070294067458238, 8.728911944873959, 9.163042147103560, 9.507208957206205, 9.957403790015003, 10.64087285629421, 10.97727205620922, 11.42573257944420, 12.06889972679673, 12.36706825344029, 12.94978800431873, 13.22767534055097
