| L(s) = 1 | + 0.794·2-s − 3-s − 1.36·4-s − 2.06·5-s − 0.794·6-s − 0.419·7-s − 2.67·8-s + 9-s − 1.64·10-s − 4.33·11-s + 1.36·12-s − 13-s − 0.333·14-s + 2.06·15-s + 0.608·16-s − 7.64·17-s + 0.794·18-s − 5.46·19-s + 2.82·20-s + 0.419·21-s − 3.44·22-s + 4.58·23-s + 2.67·24-s − 0.729·25-s − 0.794·26-s − 27-s + 0.573·28-s + ⋯ |
| L(s) = 1 | + 0.562·2-s − 0.577·3-s − 0.684·4-s − 0.924·5-s − 0.324·6-s − 0.158·7-s − 0.946·8-s + 0.333·9-s − 0.519·10-s − 1.30·11-s + 0.394·12-s − 0.277·13-s − 0.0890·14-s + 0.533·15-s + 0.152·16-s − 1.85·17-s + 0.187·18-s − 1.25·19-s + 0.632·20-s + 0.0914·21-s − 0.734·22-s + 0.955·23-s + 0.546·24-s − 0.145·25-s − 0.155·26-s − 0.192·27-s + 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09840871334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09840871334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.794T + 2T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 + 0.419T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 17 | \( 1 + 7.64T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 4.87T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 + 8.00T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 7.89T + 67T^{2} \) |
| 71 | \( 1 - 4.14T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.01T + 83T^{2} \) |
| 89 | \( 1 - 6.93T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.560904394591357711361790577911, −7.64414129140202640325435996516, −6.92461531838284021213258367885, −6.11756525495691727426691706344, −5.25425747378383163429143807986, −4.66260893881651936542781168818, −4.08739574891244369198276030179, −3.19470928534644292873982008898, −2.12497763271320951455026078728, −0.15891806735839184642131842264,
0.15891806735839184642131842264, 2.12497763271320951455026078728, 3.19470928534644292873982008898, 4.08739574891244369198276030179, 4.66260893881651936542781168818, 5.25425747378383163429143807986, 6.11756525495691727426691706344, 6.92461531838284021213258367885, 7.64414129140202640325435996516, 8.560904394591357711361790577911
