L(s) = 1 | + 2-s + 2.87·3-s + 4-s + 5-s + 2.87·6-s − 0.365·7-s + 8-s + 5.24·9-s + 10-s + 6.48·11-s + 2.87·12-s + 1.42·13-s − 0.365·14-s + 2.87·15-s + 16-s − 4.88·17-s + 5.24·18-s + 1.41·19-s + 20-s − 1.04·21-s + 6.48·22-s + 1.65·23-s + 2.87·24-s + 25-s + 1.42·26-s + 6.45·27-s − 0.365·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s + 0.447·5-s + 1.17·6-s − 0.138·7-s + 0.353·8-s + 1.74·9-s + 0.316·10-s + 1.95·11-s + 0.829·12-s + 0.394·13-s − 0.0976·14-s + 0.741·15-s + 0.250·16-s − 1.18·17-s + 1.23·18-s + 0.325·19-s + 0.223·20-s − 0.229·21-s + 1.38·22-s + 0.344·23-s + 0.586·24-s + 0.200·25-s + 0.279·26-s + 1.24·27-s − 0.0690·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.385267450\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.385267450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 7 | \( 1 + 0.365T + 7T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 + 3.85T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 3.84T + 71T^{2} \) |
| 73 | \( 1 + 5.54T + 73T^{2} \) |
| 79 | \( 1 + 5.56T + 79T^{2} \) |
| 83 | \( 1 - 7.13T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 6.98T + 97T^{2} \) |
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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
−8.286489145315639729014084986691, −7.23937759245640004073616829665, −6.59997609715226437670864177398, −6.25265287634185325756134415011, −4.87795539200175118614482630722, −4.30983031143051874996645995478, −3.43947620292113938086116370794, −3.07480738911970440886910507896, −1.93332921679485973810529440196, −1.42722630817379123786420688981,
1.42722630817379123786420688981, 1.93332921679485973810529440196, 3.07480738911970440886910507896, 3.43947620292113938086116370794, 4.30983031143051874996645995478, 4.87795539200175118614482630722, 6.25265287634185325756134415011, 6.59997609715226437670864177398, 7.23937759245640004073616829665, 8.286489145315639729014084986691