| L(s) = 1 | − 2.39·2-s + 3.71·4-s + 2.58·5-s + 7-s − 4.09·8-s − 6.19·10-s − 6.78·13-s − 2.39·14-s + 2.36·16-s − 3.14·17-s + 1.04·19-s + 9.61·20-s + 6.52·23-s + 1.70·25-s + 16.2·26-s + 3.71·28-s − 0.607·29-s − 8.83·31-s + 2.54·32-s + 7.51·34-s + 2.58·35-s + 8.94·37-s − 2.49·38-s − 10.6·40-s + 8.69·41-s − 4.48·43-s − 15.6·46-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.85·4-s + 1.15·5-s + 0.377·7-s − 1.44·8-s − 1.95·10-s − 1.88·13-s − 0.638·14-s + 0.590·16-s − 0.762·17-s + 0.239·19-s + 2.15·20-s + 1.36·23-s + 0.341·25-s + 3.17·26-s + 0.701·28-s − 0.112·29-s − 1.58·31-s + 0.449·32-s + 1.28·34-s + 0.437·35-s + 1.47·37-s − 0.405·38-s − 1.67·40-s + 1.35·41-s − 0.683·43-s − 2.30·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9046309990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9046309990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 0.607T + 29T^{2} \) |
| 31 | \( 1 + 8.83T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 - 0.174T + 59T^{2} \) |
| 61 | \( 1 + 8.13T + 61T^{2} \) |
| 67 | \( 1 + 7.33T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 0.589T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.70263598469090574967527248718, −7.52400403578341343171079036210, −6.73541442759570634283259797407, −6.04744811413946502377735134264, −5.15627327738166369603904010060, −4.54496004859240786454342065767, −2.99214437919731132254633833063, −2.24605934580626206109942644491, −1.74898812712152215982925946153, −0.60466661847827539411606027653,
0.60466661847827539411606027653, 1.74898812712152215982925946153, 2.24605934580626206109942644491, 2.99214437919731132254633833063, 4.54496004859240786454342065767, 5.15627327738166369603904010060, 6.04744811413946502377735134264, 6.73541442759570634283259797407, 7.52400403578341343171079036210, 7.70263598469090574967527248718
