| L(s) = 1 | + 2.47·2-s + 4.11·4-s + 2.58·5-s + 7-s + 5.22·8-s + 6.39·10-s + 5.87·13-s + 2.47·14-s + 4.70·16-s + 7.51·17-s + 2.35·19-s + 10.6·20-s − 6.94·23-s + 1.69·25-s + 14.5·26-s + 4.11·28-s − 5.87·29-s − 3.66·31-s + 1.16·32-s + 18.5·34-s + 2.58·35-s + 3.30·37-s + 5.83·38-s + 13.5·40-s + 5.28·41-s − 7.40·43-s − 17.1·46-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.05·4-s + 1.15·5-s + 0.377·7-s + 1.84·8-s + 2.02·10-s + 1.62·13-s + 0.660·14-s + 1.17·16-s + 1.82·17-s + 0.540·19-s + 2.38·20-s − 1.44·23-s + 0.339·25-s + 2.84·26-s + 0.777·28-s − 1.09·29-s − 0.657·31-s + 0.206·32-s + 3.18·34-s + 0.437·35-s + 0.543·37-s + 0.945·38-s + 2.13·40-s + 0.825·41-s − 1.12·43-s − 2.53·46-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
| \(L(1)\) |
\(\approx\) |
\(9.265284299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.265284299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 0.926T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.69472291003485105731155528866, −6.86053027695925187885525982916, −5.93162407070631273236998177169, −5.80521282517217095017927290241, −5.30111751047685863997206755598, −4.25109106993133927387803178021, −3.62485896320155392971296729831, −2.99228654683653474208806287522, −1.89872188842141517084720469470, −1.38966823335737992238781390990,
1.38966823335737992238781390990, 1.89872188842141517084720469470, 2.99228654683653474208806287522, 3.62485896320155392971296729831, 4.25109106993133927387803178021, 5.30111751047685863997206755598, 5.80521282517217095017927290241, 5.93162407070631273236998177169, 6.86053027695925187885525982916, 7.69472291003485105731155528866
