| L(s) = 1 | + 1.58·2-s + 3-s + 0.516·4-s − 3.36·5-s + 1.58·6-s − 2.35·8-s + 9-s − 5.34·10-s − 2.28·11-s + 0.516·12-s − 5.46·13-s − 3.36·15-s − 4.76·16-s − 0.821·17-s + 1.58·18-s − 6.52·19-s − 1.73·20-s − 3.61·22-s − 3.49·23-s − 2.35·24-s + 6.34·25-s − 8.66·26-s + 27-s + 3.18·29-s − 5.34·30-s − 0.597·31-s − 2.85·32-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.577·3-s + 0.258·4-s − 1.50·5-s + 0.647·6-s − 0.832·8-s + 0.333·9-s − 1.68·10-s − 0.687·11-s + 0.149·12-s − 1.51·13-s − 0.869·15-s − 1.19·16-s − 0.199·17-s + 0.373·18-s − 1.49·19-s − 0.388·20-s − 0.771·22-s − 0.727·23-s − 0.480·24-s + 1.26·25-s − 1.70·26-s + 0.192·27-s + 0.591·29-s − 0.975·30-s − 0.107·31-s − 0.504·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.324562663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324562663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 0.821T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 + 0.597T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 - 9.93T + 41T^{2} \) |
| 43 | \( 1 - 0.228T + 43T^{2} \) |
| 47 | \( 1 - 0.251T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 + 5.92T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 3.74T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 - 5.99T + 97T^{2} \) |
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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
−7.68165371505755284928614359090, −7.18020434888736434442173052900, −6.35580890990164428039548788525, −5.49152531304532561165079924389, −4.62088735281885196710643650773, −4.28224405130490536779080500898, −3.74259066792334512214083758320, −2.69046412250578680517705126265, −2.39524640520184914165336908127, −0.42500247561713671084382134979,
0.42500247561713671084382134979, 2.39524640520184914165336908127, 2.69046412250578680517705126265, 3.74259066792334512214083758320, 4.28224405130490536779080500898, 4.62088735281885196710643650773, 5.49152531304532561165079924389, 6.35580890990164428039548788525, 7.18020434888736434442173052900, 7.68165371505755284928614359090
