| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s − 3·9-s + 2·10-s − 4·11-s − 4·13-s + 14-s + 16-s − 3·18-s + 6·19-s + 2·20-s − 4·22-s + 23-s − 25-s − 4·26-s + 28-s − 10·31-s + 32-s + 2·35-s − 3·36-s + 10·37-s + 6·38-s + 2·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s − 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.707·18-s + 1.37·19-s + 0.447·20-s − 0.852·22-s + 0.208·23-s − 1/5·25-s − 0.784·26-s + 0.188·28-s − 1.79·31-s + 0.176·32-s + 0.338·35-s − 1/2·36-s + 1.64·37-s + 0.973·38-s + 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.058434580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058434580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.86483683633013, −12.27157156929908, −12.05688585058587, −11.23889664639774, −11.12817056331441, −10.62687397050409, −9.950632595165978, −9.526372716378957, −9.345686870982873, −8.498725322911308, −7.961468007977065, −7.609103001372690, −7.209741917982990, −6.491091750961606, −5.958916197467340, −5.497426895892217, −5.175841911145022, −4.905726648621536, −4.134928548186645, −3.391196677862017, −2.816118775689735, −2.610449876378022, −1.891660819743130, −1.393938101885629, −0.3106013545766683,
0.3106013545766683, 1.393938101885629, 1.891660819743130, 2.610449876378022, 2.816118775689735, 3.391196677862017, 4.134928548186645, 4.905726648621536, 5.175841911145022, 5.497426895892217, 5.958916197467340, 6.491091750961606, 7.209741917982990, 7.609103001372690, 7.961468007977065, 8.498725322911308, 9.345686870982873, 9.526372716378957, 9.950632595165978, 10.62687397050409, 11.12817056331441, 11.23889664639774, 12.05688585058587, 12.27157156929908, 12.86483683633013
