| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s + 4·11-s + 12-s − 13-s + 14-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s + 24-s − 25-s − 26-s + 27-s + 28-s − 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.974238001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.974238001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.43385970521712, −14.71984550931870, −14.43050904382491, −13.84330311250612, −13.56317769303166, −12.93820443862305, −12.35630861354735, −11.73531704475622, −11.24312190733125, −10.68505194403982, −9.782104224501862, −9.464597703833527, −9.046997414518455, −8.163315729097514, −7.551711143404256, −7.036548756240527, −6.206709659071709, −5.920265879940151, −5.057620545897043, −4.501175132714805, −3.769839489335984, −3.202445303776933, −2.234400620063044, −1.844310233852938, −0.9604526149120788,
0.9604526149120788, 1.844310233852938, 2.234400620063044, 3.202445303776933, 3.769839489335984, 4.501175132714805, 5.057620545897043, 5.920265879940151, 6.206709659071709, 7.036548756240527, 7.551711143404256, 8.163315729097514, 9.046997414518455, 9.464597703833527, 9.782104224501862, 10.68505194403982, 11.24312190733125, 11.73531704475622, 12.35630861354735, 12.93820443862305, 13.56317769303166, 13.84330311250612, 14.43050904382491, 14.71984550931870, 15.43385970521712
