L(s) = 1 | + 2-s + 3-s + 4-s + 0.0205·5-s + 6-s + 5.09·7-s + 8-s + 9-s + 0.0205·10-s − 4.04·11-s + 12-s − 2.31·13-s + 5.09·14-s + 0.0205·15-s + 16-s + 17-s + 18-s + 5.61·19-s + 0.0205·20-s + 5.09·21-s − 4.04·22-s + 7.47·23-s + 24-s − 4.99·25-s − 2.31·26-s + 27-s + 5.09·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.00918·5-s + 0.408·6-s + 1.92·7-s + 0.353·8-s + 0.333·9-s + 0.00649·10-s − 1.21·11-s + 0.288·12-s − 0.642·13-s + 1.36·14-s + 0.00530·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.28·19-s + 0.00459·20-s + 1.11·21-s − 0.862·22-s + 1.55·23-s + 0.204·24-s − 0.999·25-s − 0.454·26-s + 0.192·27-s + 0.961·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.099985156\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.099985156\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 0.0205T + 5T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 19 | \( 1 - 5.61T + 19T^{2} \) |
| 23 | \( 1 - 7.47T + 23T^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 8.15T + 37T^{2} \) |
| 41 | \( 1 - 3.48T + 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 - 9.93T + 67T^{2} \) |
| 71 | \( 1 - 1.56T + 71T^{2} \) |
| 73 | \( 1 - 6.72T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 0.494T + 97T^{2} \) |
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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
−7.87906638032259721805736445888, −7.55249562260217540722225790626, −6.84053638841717322178535533549, −5.52401864085371425354118452712, −5.07162809466553897538046136116, −4.71190631924460019156714491657, −3.61748842758742704647575249684, −2.77033343655859025380812240490, −2.06020045271414231910368120935, −1.12096917291749051955579087840,
1.12096917291749051955579087840, 2.06020045271414231910368120935, 2.77033343655859025380812240490, 3.61748842758742704647575249684, 4.71190631924460019156714491657, 5.07162809466553897538046136116, 5.52401864085371425354118452712, 6.84053638841717322178535533549, 7.55249562260217540722225790626, 7.87906638032259721805736445888