L(s) = 1 | + 0.756·2-s − 1.42·4-s − 0.386·5-s + 0.194·7-s − 2.59·8-s − 0.292·10-s − 2.36·11-s − 1.22·13-s + 0.147·14-s + 0.894·16-s + 5.04·17-s + 4.24·19-s + 0.551·20-s − 1.78·22-s − 1.53·23-s − 4.85·25-s − 0.927·26-s − 0.278·28-s + 7.31·29-s + 3.33·31-s + 5.86·32-s + 3.81·34-s − 0.0752·35-s − 2.17·37-s + 3.21·38-s + 1.00·40-s + 1.04·41-s + ⋯ |
L(s) = 1 | + 0.534·2-s − 0.713·4-s − 0.172·5-s + 0.0736·7-s − 0.916·8-s − 0.0923·10-s − 0.712·11-s − 0.340·13-s + 0.0394·14-s + 0.223·16-s + 1.22·17-s + 0.973·19-s + 0.123·20-s − 0.381·22-s − 0.319·23-s − 0.970·25-s − 0.181·26-s − 0.0525·28-s + 1.35·29-s + 0.599·31-s + 1.03·32-s + 0.654·34-s − 0.0127·35-s − 0.356·37-s + 0.520·38-s + 0.158·40-s + 0.163·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 2 | \( 1 - 0.756T + 2T^{2} \) |
| 5 | \( 1 + 0.386T + 5T^{2} \) |
| 7 | \( 1 - 0.194T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.902T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.964232723671993433684856091282, −7.41629889863194649271442882666, −6.27547449401793477163562007598, −5.62395847851031756586199882705, −4.96030408524058922666475114712, −4.32971240787870679568280130113, −3.33215868831199696653215634328, −2.80373902542941369447056359433, −1.30515919019232183827556215269, 0,
1.30515919019232183827556215269, 2.80373902542941369447056359433, 3.33215868831199696653215634328, 4.32971240787870679568280130113, 4.96030408524058922666475114712, 5.62395847851031756586199882705, 6.27547449401793477163562007598, 7.41629889863194649271442882666, 7.964232723671993433684856091282