L(s) = 1 | + 0.690·2-s − 3·3-s − 7.52·4-s + 4.55·5-s − 2.07·6-s + 26.5·7-s − 10.7·8-s + 9·9-s + 3.14·10-s + 11·11-s + 22.5·12-s + 14.5·13-s + 18.2·14-s − 13.6·15-s + 52.7·16-s + 68.2·17-s + 6.21·18-s + 139.·19-s − 34.2·20-s − 79.5·21-s + 7.59·22-s + 17.3·23-s + 32.1·24-s − 104.·25-s + 10.0·26-s − 27·27-s − 199.·28-s + ⋯ |
L(s) = 1 | + 0.243·2-s − 0.577·3-s − 0.940·4-s + 0.407·5-s − 0.140·6-s + 1.43·7-s − 0.473·8-s + 0.333·9-s + 0.0994·10-s + 0.301·11-s + 0.542·12-s + 0.310·13-s + 0.349·14-s − 0.235·15-s + 0.824·16-s + 0.973·17-s + 0.0813·18-s + 1.68·19-s − 0.383·20-s − 0.826·21-s + 0.0735·22-s + 0.157·23-s + 0.273·24-s − 0.833·25-s + 0.0757·26-s − 0.192·27-s − 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.551751385\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.551751385\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
good | 2 | \( 1 - 0.690T + 8T^{2} \) |
| 5 | \( 1 - 4.55T + 125T^{2} \) |
| 7 | \( 1 - 26.5T + 343T^{2} \) |
| 13 | \( 1 - 14.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 17.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 316.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 131.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 261.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 996.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 53.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 143.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.45e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 987.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−8.867747591170626114938762238248, −7.84620747063671053718883929221, −7.52046572305246120470979030937, −5.92222820027727232901407360420, −5.67478425317837964895384730603, −4.75157438789250417745740782790, −4.17092213177188251427411013014, −3.01888184084603836814253847234, −1.49521096992562817880073322856, −0.836913857490103003999332175457,
0.836913857490103003999332175457, 1.49521096992562817880073322856, 3.01888184084603836814253847234, 4.17092213177188251427411013014, 4.75157438789250417745740782790, 5.67478425317837964895384730603, 5.92222820027727232901407360420, 7.52046572305246120470979030937, 7.84620747063671053718883929221, 8.867747591170626114938762238248