| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s − 2·9-s + 10-s + 4·11-s − 12-s + 2·13-s − 2·14-s + 15-s + 16-s + 7·17-s + 2·18-s − 5·19-s − 20-s − 2·21-s − 4·22-s − 2·23-s + 24-s + 25-s − 2·26-s + 5·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s + 0.471·18-s − 1.14·19-s − 0.223·20-s − 0.436·21-s − 0.852·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.962·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.22480595217057, −13.61105055154371, −12.76287698448411, −12.28614852247523, −11.95109957140252, −11.49813198404824, −11.05296620833929, −10.68634370187652, −10.17651074387536, −9.409009885522439, −9.075984024939647, −8.443321747761699, −8.057253178445175, −7.671610848222646, −6.946357471936389, −6.321164247343229, −6.064042996948497, −5.376831519745937, −4.826371513239207, −4.120709828123668, −3.493896304130724, −3.055595206176432, −1.971817430471787, −1.507857879066133, −0.8232833084693176, 0,
0.8232833084693176, 1.507857879066133, 1.971817430471787, 3.055595206176432, 3.493896304130724, 4.120709828123668, 4.826371513239207, 5.376831519745937, 6.064042996948497, 6.321164247343229, 6.946357471936389, 7.671610848222646, 8.057253178445175, 8.443321747761699, 9.075984024939647, 9.409009885522439, 10.17651074387536, 10.68634370187652, 11.05296620833929, 11.49813198404824, 11.95109957140252, 12.28614852247523, 12.76287698448411, 13.61105055154371, 14.22480595217057
