| L(s) = 1 | − 4·2-s + 2·3-s + 8·4-s − 5·5-s − 8·6-s + 6·7-s − 23·9-s + 20·10-s + 32·11-s + 16·12-s − 38·13-s − 24·14-s − 10·15-s − 64·16-s + 26·17-s + 92·18-s + 100·19-s − 40·20-s + 12·21-s − 128·22-s − 78·23-s + 25·25-s + 152·26-s − 100·27-s + 48·28-s − 50·29-s + 40·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.384·3-s + 4-s − 0.447·5-s − 0.544·6-s + 0.323·7-s − 0.851·9-s + 0.632·10-s + 0.877·11-s + 0.384·12-s − 0.810·13-s − 0.458·14-s − 0.172·15-s − 16-s + 0.370·17-s + 1.20·18-s + 1.20·19-s − 0.447·20-s + 0.124·21-s − 1.24·22-s − 0.707·23-s + 1/5·25-s + 1.14·26-s − 0.712·27-s + 0.323·28-s − 0.320·29-s + 0.243·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4118613283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4118613283\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 26 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 442 T + p^{3} T^{2} \) |
| 47 | \( 1 + 514 T + p^{3} T^{2} \) |
| 53 | \( 1 - 2 T + p^{3} T^{2} \) |
| 59 | \( 1 - 500 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 126 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 878 T + p^{3} T^{2} \) |
| 79 | \( 1 - 600 T + p^{3} T^{2} \) |
| 83 | \( 1 - 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 150 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−24.53719930434053859680587393842, −22.51370164112999450106246009488, −20.24253638460194980453802000495, −19.39406085946034311371495394382, −17.80936002011203375803360468238, −16.49202096766309066044503764833, −14.42700262096538081523197027989, −11.52156568135998034842009582479, −9.415016459828749011981823541780, −7.80368599340441310768273857061,
7.80368599340441310768273857061, 9.415016459828749011981823541780, 11.52156568135998034842009582479, 14.42700262096538081523197027989, 16.49202096766309066044503764833, 17.80936002011203375803360468238, 19.39406085946034311371495394382, 20.24253638460194980453802000495, 22.51370164112999450106246009488, 24.53719930434053859680587393842
