L(s) = 1 | + 2-s + 0.625·3-s + 4-s − 0.793·5-s + 0.625·6-s − 2.19·7-s + 8-s − 2.60·9-s − 0.793·10-s + 1.89·11-s + 0.625·12-s + 7.04·13-s − 2.19·14-s − 0.496·15-s + 16-s + 7.96·17-s − 2.60·18-s − 19-s − 0.793·20-s − 1.37·21-s + 1.89·22-s + 1.50·23-s + 0.625·24-s − 4.37·25-s + 7.04·26-s − 3.50·27-s − 2.19·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.361·3-s + 0.5·4-s − 0.354·5-s + 0.255·6-s − 0.829·7-s + 0.353·8-s − 0.869·9-s − 0.250·10-s + 0.570·11-s + 0.180·12-s + 1.95·13-s − 0.586·14-s − 0.128·15-s + 0.250·16-s + 1.93·17-s − 0.614·18-s − 0.229·19-s − 0.177·20-s − 0.299·21-s + 0.403·22-s + 0.314·23-s + 0.127·24-s − 0.874·25-s + 1.38·26-s − 0.675·27-s − 0.414·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.461227469\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.461227469\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 0.625T + 3T^{2} \) |
| 5 | \( 1 + 0.793T + 5T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 13 | \( 1 - 7.04T + 13T^{2} \) |
| 17 | \( 1 - 7.96T + 17T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 - 5.10T + 61T^{2} \) |
| 67 | \( 1 - 3.86T + 67T^{2} \) |
| 71 | \( 1 - 5.10T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 + 2.03T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.78333647631646844402511583358, −7.09820891273657583996297815587, −6.16157782597415491023750438990, −5.90069675519680186458644428411, −5.16574917314931732240619924494, −3.78065037787998933836566097494, −3.67181125349319172097914328257, −3.07077015333357996717586987227, −1.89797304772003857429221914576, −0.826155290570608958576189721809,
0.826155290570608958576189721809, 1.89797304772003857429221914576, 3.07077015333357996717586987227, 3.67181125349319172097914328257, 3.78065037787998933836566097494, 5.16574917314931732240619924494, 5.90069675519680186458644428411, 6.16157782597415491023750438990, 7.09820891273657583996297815587, 7.78333647631646844402511583358