| L(s) = 1 | + 2.58·2-s + 3.12·3-s + 4.67·4-s − 4.14·5-s + 8.06·6-s − 3.36·7-s + 6.90·8-s + 6.73·9-s − 10.7·10-s + 2.17·11-s + 14.5·12-s − 2.33·13-s − 8.70·14-s − 12.9·15-s + 8.48·16-s + 0.306·17-s + 17.3·18-s − 7.75·19-s − 19.3·20-s − 10.5·21-s + 5.61·22-s + 1.14·23-s + 21.5·24-s + 12.2·25-s − 6.03·26-s + 11.6·27-s − 15.7·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 1.80·3-s + 2.33·4-s − 1.85·5-s + 3.29·6-s − 1.27·7-s + 2.44·8-s + 2.24·9-s − 3.38·10-s + 0.655·11-s + 4.20·12-s − 0.648·13-s − 2.32·14-s − 3.34·15-s + 2.12·16-s + 0.0742·17-s + 4.10·18-s − 1.77·19-s − 4.33·20-s − 2.29·21-s + 1.19·22-s + 0.239·23-s + 4.39·24-s + 2.44·25-s − 1.18·26-s + 2.24·27-s − 2.97·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.832323675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.832323675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 - 0.306T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 + 2.13T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 - 0.876T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 1.93T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 4.79T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 - 0.0400T + 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−11.31172498155604783739946644866, −10.13615844905431108245185816867, −8.910946338977898986257951557091, −8.006036278985761105062348092767, −7.10333630988646437317547331464, −6.53908669696827806762431878039, −4.55894496207705865729085149794, −3.92714273462630635595210706716, −3.34105070557092149436634715545, −2.47856230293590711538130151604,
2.47856230293590711538130151604, 3.34105070557092149436634715545, 3.92714273462630635595210706716, 4.55894496207705865729085149794, 6.53908669696827806762431878039, 7.10333630988646437317547331464, 8.006036278985761105062348092767, 8.910946338977898986257951557091, 10.13615844905431108245185816867, 11.31172498155604783739946644866
