| L(s) = 1 | − 3.10·3-s − 2.03·5-s + 0.883·7-s + 6.64·9-s − 4.95·11-s + 5.43·13-s + 6.31·15-s + 1.35·19-s − 2.74·21-s + 4.73·23-s − 0.863·25-s − 11.3·27-s − 3.53·29-s − 8.64·31-s + 15.4·33-s − 1.79·35-s − 5.77·37-s − 16.8·39-s + 10.6·41-s − 11.4·43-s − 13.5·45-s + 6.32·47-s − 6.21·49-s + 3.44·53-s + 10.0·55-s − 4.21·57-s − 0.528·59-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.909·5-s + 0.333·7-s + 2.21·9-s − 1.49·11-s + 1.50·13-s + 1.63·15-s + 0.311·19-s − 0.598·21-s + 0.986·23-s − 0.172·25-s − 2.17·27-s − 0.655·29-s − 1.55·31-s + 2.68·33-s − 0.303·35-s − 0.949·37-s − 2.70·39-s + 1.66·41-s − 1.74·43-s − 2.01·45-s + 0.922·47-s − 0.888·49-s + 0.472·53-s + 1.36·55-s − 0.557·57-s − 0.0688·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 - 0.883T + 7T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 + 5.77T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 + 0.528T + 59T^{2} \) |
| 61 | \( 1 - 9.54T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 - 8.22T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 2.79T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.24939757092923412832639452004, −6.73198337717369302197780211720, −5.69185123479958036625318139684, −5.49879201626795041803064902519, −4.77774834630919133709631266735, −3.97830535345167066394519868636, −3.31635369144868361002705648449, −1.89601859842863580717662055104, −0.878624953432363225882843095122, 0,
0.878624953432363225882843095122, 1.89601859842863580717662055104, 3.31635369144868361002705648449, 3.97830535345167066394519868636, 4.77774834630919133709631266735, 5.49879201626795041803064902519, 5.69185123479958036625318139684, 6.73198337717369302197780211720, 7.24939757092923412832639452004
