| L(s) = 1 | − 2-s + 2.57·3-s + 4-s − 5-s − 2.57·6-s − 1.90·7-s − 8-s + 3.61·9-s + 10-s − 5.17·11-s + 2.57·12-s + 1.90·14-s − 2.57·15-s + 16-s − 5.89·17-s − 3.61·18-s + 8.09·19-s − 20-s − 4.89·21-s + 5.17·22-s − 0.908·23-s − 2.57·24-s + 25-s + 1.58·27-s − 1.90·28-s − 5.29·29-s + 2.57·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.48·3-s + 0.5·4-s − 0.447·5-s − 1.05·6-s − 0.719·7-s − 0.353·8-s + 1.20·9-s + 0.316·10-s − 1.56·11-s + 0.742·12-s + 0.508·14-s − 0.664·15-s + 0.250·16-s − 1.42·17-s − 0.852·18-s + 1.85·19-s − 0.223·20-s − 1.06·21-s + 1.10·22-s − 0.189·23-s − 0.525·24-s + 0.200·25-s + 0.305·27-s − 0.359·28-s − 0.984·29-s + 0.469·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 - 8.09T + 19T^{2} \) |
| 23 | \( 1 + 0.908T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 0.0626T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 0.681T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 0.163T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 1.89T + 83T^{2} \) |
| 89 | \( 1 - 0.541T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−9.011096678555291283921050979241, −8.157494676004059019323248995367, −7.59324740956256258052955787569, −7.04860534618231122369362412768, −5.81519431204441667872574491247, −4.64777491221740254210430803276, −3.34766247948660079958349368348, −2.92309136413931017118067807527, −1.88674933520608226508005298979, 0,
1.88674933520608226508005298979, 2.92309136413931017118067807527, 3.34766247948660079958349368348, 4.64777491221740254210430803276, 5.81519431204441667872574491247, 7.04860534618231122369362412768, 7.59324740956256258052955787569, 8.157494676004059019323248995367, 9.011096678555291283921050979241
