L(s) = 1 | − 6.05·3-s − 8.16·5-s − 11.0·7-s + 9.61·9-s + 26.1·11-s − 34.2·13-s + 49.4·15-s − 130.·19-s + 66.6·21-s − 74.8·23-s − 58.2·25-s + 105.·27-s + 56.8·29-s − 90.6·31-s − 158.·33-s + 89.9·35-s − 206.·37-s + 207.·39-s + 171.·41-s − 169.·43-s − 78.4·45-s + 154.·47-s − 221.·49-s − 281.·53-s − 213.·55-s + 787.·57-s − 448.·59-s + ⋯ |
L(s) = 1 | − 1.16·3-s − 0.730·5-s − 0.594·7-s + 0.355·9-s + 0.716·11-s − 0.729·13-s + 0.850·15-s − 1.57·19-s + 0.692·21-s − 0.678·23-s − 0.466·25-s + 0.749·27-s + 0.363·29-s − 0.525·31-s − 0.833·33-s + 0.434·35-s − 0.918·37-s + 0.849·39-s + 0.651·41-s − 0.601·43-s − 0.260·45-s + 0.479·47-s − 0.646·49-s − 0.728·53-s − 0.523·55-s + 1.83·57-s − 0.989·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0001145920850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0001145920850\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 6.05T + 27T^{2} \) |
| 5 | \( 1 + 8.16T + 125T^{2} \) |
| 7 | \( 1 + 11.0T + 343T^{2} \) |
| 11 | \( 1 - 26.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 56.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 154.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 281.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 448.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 910.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 661.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 135.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 776.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−8.645290409519547686013782715670, −7.79622938264079192698407924659, −6.89100541746581445605668508466, −6.32801561927970816320726184229, −5.64048491210804854848063511117, −4.58837342224915350175048523434, −4.01933241660223928728202896423, −2.90776616642005814796343646905, −1.59545157147313305856480454404, −0.00502832029547353218602033198,
0.00502832029547353218602033198, 1.59545157147313305856480454404, 2.90776616642005814796343646905, 4.01933241660223928728202896423, 4.58837342224915350175048523434, 5.64048491210804854848063511117, 6.32801561927970816320726184229, 6.89100541746581445605668508466, 7.79622938264079192698407924659, 8.645290409519547686013782715670