L(s) = 1 | + 2·2-s + 8.92·3-s + 4·4-s − 5·5-s + 17.8·6-s + 23.5·7-s + 8·8-s + 52.7·9-s − 10·10-s + 1.63·11-s + 35.7·12-s − 88.6·13-s + 47.1·14-s − 44.6·15-s + 16·16-s − 7.31·17-s + 105.·18-s − 6.53·19-s − 20·20-s + 210.·21-s + 3.27·22-s + 23·23-s + 71.4·24-s + 25·25-s − 177.·26-s + 229.·27-s + 94.2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s − 0.447·5-s + 1.21·6-s + 1.27·7-s + 0.353·8-s + 1.95·9-s − 0.316·10-s + 0.0448·11-s + 0.859·12-s − 1.89·13-s + 0.899·14-s − 0.768·15-s + 0.250·16-s − 0.104·17-s + 1.38·18-s − 0.0789·19-s − 0.223·20-s + 2.18·21-s + 0.0316·22-s + 0.208·23-s + 0.607·24-s + 0.200·25-s − 1.33·26-s + 1.63·27-s + 0.635·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.716132253\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.716132253\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
good | 3 | \( 1 - 8.92T + 27T^{2} \) |
| 7 | \( 1 - 23.5T + 343T^{2} \) |
| 11 | \( 1 - 1.63T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.31T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.53T + 6.85e3T^{2} \) |
| 29 | \( 1 - 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 74.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 233.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 766.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 178.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 246.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 650.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 695.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 660.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 824.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 383.T + 9.12e5T^{2} \) |
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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
−12.04899734650032231224440300505, −10.85508026848754348364847229614, −9.685042683356738689637119605035, −8.612880457287322246343734423696, −7.72764494713986537189133122686, −7.13817456767071836379134482158, −5.04514031914002366085434525712, −4.20802066450498989423814407225, −2.89640745608174341720521656897, −1.87860545020693535218493606153,
1.87860545020693535218493606153, 2.89640745608174341720521656897, 4.20802066450498989423814407225, 5.04514031914002366085434525712, 7.13817456767071836379134482158, 7.72764494713986537189133122686, 8.612880457287322246343734423696, 9.685042683356738689637119605035, 10.85508026848754348364847229614, 12.04899734650032231224440300505