| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 11-s + 12-s + 13-s + 16-s + 17-s + 18-s − 4·19-s − 22-s + 24-s + 26-s + 27-s − 2·29-s − 8·31-s + 32-s − 33-s + 34-s + 36-s − 2·37-s − 4·38-s + 39-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.213·22-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.160·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.73501504296663, −12.37503942079409, −12.06218482881566, −11.33554813286716, −10.91759200566282, −10.60139804469598, −10.14089145083099, −9.416002760754465, −9.220394071437184, −8.538942080197526, −8.129202507034599, −7.746165902802106, −7.066450989938747, −6.839815417499323, −6.200378065305408, −5.697341530480838, −5.210757859264419, −4.770928965337222, −4.082419878415474, −3.711811048616420, −3.306777616773749, −2.631223067141034, −2.075419507351509, −1.715121562422015, −0.8704526984085200, 0,
0.8704526984085200, 1.715121562422015, 2.075419507351509, 2.631223067141034, 3.306777616773749, 3.711811048616420, 4.082419878415474, 4.770928965337222, 5.210757859264419, 5.697341530480838, 6.200378065305408, 6.839815417499323, 7.066450989938747, 7.746165902802106, 8.129202507034599, 8.538942080197526, 9.220394071437184, 9.416002760754465, 10.14089145083099, 10.60139804469598, 10.91759200566282, 11.33554813286716, 12.06218482881566, 12.37503942079409, 12.73501504296663
