| L(s) = 1 | + 0.342·2-s − 2.49·3-s − 1.88·4-s + 4.14·5-s − 0.853·6-s − 0.810·7-s − 1.32·8-s + 3.21·9-s + 1.41·10-s − 1.97·11-s + 4.69·12-s − 0.639·13-s − 0.277·14-s − 10.3·15-s + 3.31·16-s + 6.07·17-s + 1.10·18-s − 4.15·19-s − 7.79·20-s + 2.02·21-s − 0.677·22-s + 7.02·23-s + 3.31·24-s + 12.1·25-s − 0.218·26-s − 0.545·27-s + 1.52·28-s + ⋯ |
| L(s) = 1 | + 0.241·2-s − 1.43·3-s − 0.941·4-s + 1.85·5-s − 0.348·6-s − 0.306·7-s − 0.469·8-s + 1.07·9-s + 0.448·10-s − 0.596·11-s + 1.35·12-s − 0.177·13-s − 0.0741·14-s − 2.66·15-s + 0.827·16-s + 1.47·17-s + 0.259·18-s − 0.953·19-s − 1.74·20-s + 0.441·21-s − 0.144·22-s + 1.46·23-s + 0.676·24-s + 2.43·25-s − 0.0429·26-s − 0.105·27-s + 0.288·28-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
| \(L(1)\) |
\(\approx\) |
\(1.040463293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040463293\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.342T + 2T^{2} \) |
| 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 + 0.810T + 7T^{2} \) |
| 11 | \( 1 + 1.97T + 11T^{2} \) |
| 13 | \( 1 + 0.639T + 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 7.02T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 9.97T + 41T^{2} \) |
| 43 | \( 1 + 7.72T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 - 2.45T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 6.33T + 71T^{2} \) |
| 73 | \( 1 + 8.43T + 73T^{2} \) |
| 79 | \( 1 - 1.49T + 79T^{2} \) |
| 83 | \( 1 - 3.66T + 83T^{2} \) |
| 89 | \( 1 - 5.96T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−10.51606059272698692628333814969, −10.27360035882437705774730397311, −9.439871220508031627752647916415, −8.427916659506060954653349229260, −6.76770828674286862357096891315, −6.02209877188187404709195491920, −5.30646160789985622576538529102, −4.78083987521309006947729940072, −2.89820161154036772676063496917, −1.01814591614460726602083782884,
1.01814591614460726602083782884, 2.89820161154036772676063496917, 4.78083987521309006947729940072, 5.30646160789985622576538529102, 6.02209877188187404709195491920, 6.76770828674286862357096891315, 8.427916659506060954653349229260, 9.439871220508031627752647916415, 10.27360035882437705774730397311, 10.51606059272698692628333814969
