| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 5·11-s − 13-s + 4·14-s + 16-s + 19-s − 20-s + 5·22-s + 3·23-s + 25-s + 26-s − 4·28-s − 9·29-s + 3·31-s − 32-s + 4·35-s − 37-s − 38-s + 40-s − 8·41-s + 11·43-s − 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 1.50·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s + 1.06·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.676·35-s − 0.164·37-s − 0.162·38-s + 0.158·40-s − 1.24·41-s + 1.67·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1200222644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1200222644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.52929367148121, −12.17704184806761, −11.67203953062434, −10.98551654543683, −10.78493793163726, −10.15903017998703, −9.945195908361203, −9.397873020965811, −8.932832174600466, −8.580776755487546, −7.874469091753488, −7.499036089533192, −7.148890929426537, −6.737784007312125, −6.012846350514916, −5.641386124556549, −5.221822106138435, −4.442914228575191, −3.899566752922714, −3.252202118023885, −2.853419230037617, −2.490870455563920, −1.705638946359795, −0.8587281010357263, −0.1253518978050891,
0.1253518978050891, 0.8587281010357263, 1.705638946359795, 2.490870455563920, 2.853419230037617, 3.252202118023885, 3.899566752922714, 4.442914228575191, 5.221822106138435, 5.641386124556549, 6.012846350514916, 6.737784007312125, 7.148890929426537, 7.499036089533192, 7.874469091753488, 8.580776755487546, 8.932832174600466, 9.397873020965811, 9.945195908361203, 10.15903017998703, 10.78493793163726, 10.98551654543683, 11.67203953062434, 12.17704184806761, 12.52929367148121
