| L(s) = 1 | − 2-s + 2.57·3-s + 4-s + 3.18·5-s − 2.57·6-s − 4.63·7-s − 8-s + 3.65·9-s − 3.18·10-s − 1.26·11-s + 2.57·12-s − 4.01·13-s + 4.63·14-s + 8.20·15-s + 16-s + 1.91·17-s − 3.65·18-s − 4.81·19-s + 3.18·20-s − 11.9·21-s + 1.26·22-s + 23-s − 2.57·24-s + 5.12·25-s + 4.01·26-s + 1.68·27-s − 4.63·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.48·3-s + 0.5·4-s + 1.42·5-s − 1.05·6-s − 1.75·7-s − 0.353·8-s + 1.21·9-s − 1.00·10-s − 0.381·11-s + 0.744·12-s − 1.11·13-s + 1.23·14-s + 2.11·15-s + 0.250·16-s + 0.465·17-s − 0.860·18-s − 1.10·19-s + 0.711·20-s − 2.60·21-s + 0.269·22-s + 0.208·23-s − 0.526·24-s + 1.02·25-s + 0.787·26-s + 0.323·27-s − 0.876·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 0.522T + 31T^{2} \) |
| 37 | \( 1 - 0.765T + 37T^{2} \) |
| 41 | \( 1 + 3.04T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 7.28T + 61T^{2} \) |
| 67 | \( 1 - 0.101T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.43T + 73T^{2} \) |
| 79 | \( 1 + 8.28T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 4.39T + 97T^{2} \) |
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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
−7.85535660873221889518468194719, −7.05539417803620660269034714086, −6.50614240632999307048938116909, −5.85890388827336730180332540744, −4.84959307689045157222316510157, −3.59133165268862119375145523672, −2.87398660836735175519754740443, −2.45207833151307921265963047858, −1.63591772496228960282424887750, 0,
1.63591772496228960282424887750, 2.45207833151307921265963047858, 2.87398660836735175519754740443, 3.59133165268862119375145523672, 4.84959307689045157222316510157, 5.85890388827336730180332540744, 6.50614240632999307048938116909, 7.05539417803620660269034714086, 7.85535660873221889518468194719
