| L(s) = 1 | − 2-s + 4-s − 1.88·5-s − 3.39·7-s − 8-s + 1.88·10-s − 5.81·11-s + 2.85·13-s + 3.39·14-s + 16-s + 1.23·17-s − 2.52·19-s − 1.88·20-s + 5.81·22-s − 5.67·23-s − 1.43·25-s − 2.85·26-s − 3.39·28-s + 1.34·29-s − 5.94·31-s − 32-s − 1.23·34-s + 6.41·35-s − 10.2·37-s + 2.52·38-s + 1.88·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.843·5-s − 1.28·7-s − 0.353·8-s + 0.596·10-s − 1.75·11-s + 0.792·13-s + 0.908·14-s + 0.250·16-s + 0.300·17-s − 0.580·19-s − 0.421·20-s + 1.23·22-s − 1.18·23-s − 0.287·25-s − 0.560·26-s − 0.642·28-s + 0.249·29-s − 1.06·31-s − 0.176·32-s − 0.212·34-s + 1.08·35-s − 1.68·37-s + 0.410·38-s + 0.298·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2466192407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2466192407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 1.88T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 + 9.86T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 5.52T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.444811085229420632146213667537, −7.62307207197832398542656208087, −7.36913771035616056562999371967, −6.14644881821247850178359812355, −5.84526741925045052653705890066, −4.59096401373204003172059459496, −3.58487096595638161525680445390, −3.01866130045936704452810239656, −1.94543818755445071592966626596, −0.29570279429190020382977894129,
0.29570279429190020382977894129, 1.94543818755445071592966626596, 3.01866130045936704452810239656, 3.58487096595638161525680445390, 4.59096401373204003172059459496, 5.84526741925045052653705890066, 6.14644881821247850178359812355, 7.36913771035616056562999371967, 7.62307207197832398542656208087, 8.444811085229420632146213667537
