| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 7-s − 2·10-s + 5·11-s − 4·13-s − 2·14-s − 4·16-s − 19-s − 2·20-s + 10·22-s − 2·23-s + 25-s − 8·26-s − 2·28-s + 29-s − 4·31-s − 8·32-s + 35-s − 8·37-s − 2·38-s − 9·41-s − 8·43-s + 10·44-s − 4·46-s + 10·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s − 0.632·10-s + 1.50·11-s − 1.10·13-s − 0.534·14-s − 16-s − 0.229·19-s − 0.447·20-s + 2.13·22-s − 0.417·23-s + 1/5·25-s − 1.56·26-s − 0.377·28-s + 0.185·29-s − 0.718·31-s − 1.41·32-s + 0.169·35-s − 1.31·37-s − 0.324·38-s − 1.40·41-s − 1.21·43-s + 1.50·44-s − 0.589·46-s + 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.368088561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.368088561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.98659708207419, −13.45976671906616, −12.80559391885517, −12.32356316781104, −12.04020020176359, −11.73499835024318, −11.16539776023772, −10.41768291987632, −10.02055487590988, −9.226849304509892, −8.965709720101497, −8.391710106977990, −7.466491094698540, −7.153161375213446, −6.478345602552847, −6.296552471160707, −5.414483191051611, −5.045827227552501, −4.421756427301458, −3.897065540222786, −3.503363075597992, −2.944294634096445, −2.147839691438359, −1.549427895687072, −0.3646768340755023,
0.3646768340755023, 1.549427895687072, 2.147839691438359, 2.944294634096445, 3.503363075597992, 3.897065540222786, 4.421756427301458, 5.045827227552501, 5.414483191051611, 6.296552471160707, 6.478345602552847, 7.153161375213446, 7.466491094698540, 8.391710106977990, 8.965709720101497, 9.226849304509892, 10.02055487590988, 10.41768291987632, 11.16539776023772, 11.73499835024318, 12.04020020176359, 12.32356316781104, 12.80559391885517, 13.45976671906616, 13.98659708207419
