| L(s) = 1 | + 2.58·2-s − 3-s + 4.70·4-s − 2.58·6-s + 7-s + 6.99·8-s + 9-s − 11-s − 4.70·12-s + 2.40·13-s + 2.58·14-s + 8.70·16-s + 3.70·17-s + 2.58·18-s + 4.15·19-s − 21-s − 2.58·22-s + 0.0692·23-s − 6.99·24-s + 6.22·26-s − 27-s + 4.70·28-s + 0.703·29-s + 5.22·31-s + 8.53·32-s + 33-s + 9.58·34-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.35·4-s − 1.05·6-s + 0.377·7-s + 2.47·8-s + 0.333·9-s − 0.301·11-s − 1.35·12-s + 0.666·13-s + 0.691·14-s + 2.17·16-s + 0.897·17-s + 0.610·18-s + 0.953·19-s − 0.218·21-s − 0.551·22-s + 0.0144·23-s − 1.42·24-s + 1.22·26-s − 0.192·27-s + 0.888·28-s + 0.130·29-s + 0.938·31-s + 1.50·32-s + 0.174·33-s + 1.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.328819869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.328819869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 13 | \( 1 - 2.40T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 0.0692T + 23T^{2} \) |
| 29 | \( 1 - 0.703T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 8.51T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 + 5.56T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−7.77538376814930068982637129118, −7.11608412617782445806877973654, −6.38113011309740284783672481793, −5.79909161765190006321758207520, −5.08777906447073400334352179151, −4.74117980759265276178196605536, −3.65123017507506454426333568442, −3.22925658062459632426823964834, −2.10245879726846936401906276243, −1.12441438760127975125955978799,
1.12441438760127975125955978799, 2.10245879726846936401906276243, 3.22925658062459632426823964834, 3.65123017507506454426333568442, 4.74117980759265276178196605536, 5.08777906447073400334352179151, 5.79909161765190006321758207520, 6.38113011309740284783672481793, 7.11608412617782445806877973654, 7.77538376814930068982637129118
