| L(s) = 1 | − 2.56·3-s + 4.20·5-s + 1.74·7-s + 3.56·9-s − 6.20·11-s − 4.82·13-s − 10.7·15-s − 5.12·17-s − 19-s − 4.46·21-s + 0.463·23-s + 12.6·25-s − 1.43·27-s − 4.82·29-s − 1.63·31-s + 15.8·33-s + 7.32·35-s − 3.63·37-s + 12.3·39-s − 5.28·41-s − 1.08·43-s + 14.9·45-s − 0.555·47-s − 3.96·49-s + 13.1·51-s − 2.65·53-s − 26.1·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 1.88·5-s + 0.658·7-s + 1.18·9-s − 1.87·11-s − 1.33·13-s − 2.78·15-s − 1.24·17-s − 0.229·19-s − 0.974·21-s + 0.0966·23-s + 2.53·25-s − 0.276·27-s − 0.896·29-s − 0.294·31-s + 2.76·33-s + 1.23·35-s − 0.598·37-s + 1.97·39-s − 0.826·41-s − 0.165·43-s + 2.23·45-s − 0.0809·47-s − 0.566·49-s + 1.83·51-s − 0.365·53-s − 3.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 4.20T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 23 | \( 1 - 0.463T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 3.63T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 + 0.555T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 9.32T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.00646T + 73T^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−9.653006622243402634824810509268, −8.621211585459531220665468839948, −7.41738538466765150508131013236, −6.62034477251348148539150467029, −5.76443235836432001301413809826, −5.07548298225404164713768069460, −4.86288702390322254773005395949, −2.57715472574476540785951773797, −1.82685326359813927730044409360, 0,
1.82685326359813927730044409360, 2.57715472574476540785951773797, 4.86288702390322254773005395949, 5.07548298225404164713768069460, 5.76443235836432001301413809826, 6.62034477251348148539150467029, 7.41738538466765150508131013236, 8.621211585459531220665468839948, 9.653006622243402634824810509268
