| L(s) = 1 | + 2-s − 2.57·3-s + 4-s − 1.32·5-s − 2.57·6-s + 8-s + 3.63·9-s − 1.32·10-s + 2.09·11-s − 2.57·12-s − 3.11·13-s + 3.41·15-s + 16-s + 6.94·17-s + 3.63·18-s − 7.76·19-s − 1.32·20-s + 2.09·22-s − 23-s − 2.57·24-s − 3.24·25-s − 3.11·26-s − 1.64·27-s + 1.15·29-s + 3.41·30-s + 8.71·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.48·3-s + 0.5·4-s − 0.593·5-s − 1.05·6-s + 0.353·8-s + 1.21·9-s − 0.419·10-s + 0.630·11-s − 0.743·12-s − 0.862·13-s + 0.882·15-s + 0.250·16-s + 1.68·17-s + 0.857·18-s − 1.78·19-s − 0.296·20-s + 0.445·22-s − 0.208·23-s − 0.525·24-s − 0.648·25-s − 0.610·26-s − 0.316·27-s + 0.215·29-s + 0.624·30-s + 1.56·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.57T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 + 3.11T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 + 7.76T + 19T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 - 8.71T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 8.36T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 + 7.41T + 71T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 + 0.797T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 6.65T + 97T^{2} \) |
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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
−8.441864846843129690427676989878, −7.63722801761846931031515353349, −6.81287763445123522308097596855, −6.15493621986919109442691192146, −5.52407644939754820041304529367, −4.59487145852641761649756827573, −4.10770907828184637439194290402, −2.89753328676207410165536233012, −1.42327246202776324736811793723, 0,
1.42327246202776324736811793723, 2.89753328676207410165536233012, 4.10770907828184637439194290402, 4.59487145852641761649756827573, 5.52407644939754820041304529367, 6.15493621986919109442691192146, 6.81287763445123522308097596855, 7.63722801761846931031515353349, 8.441864846843129690427676989878
