| L(s) = 1 | − 3.03·3-s − 3.82·5-s − 7-s + 6.18·9-s + 5.96·11-s + 1.44·13-s + 11.6·15-s + 6.06·17-s + 0.0743·19-s + 3.03·21-s + 4.43·23-s + 9.64·25-s − 9.66·27-s − 1.92·29-s − 1.76·31-s − 18.0·33-s + 3.82·35-s + 0.497·37-s − 4.38·39-s − 41-s − 4.10·43-s − 23.6·45-s + 2.92·47-s + 49-s − 18.3·51-s + 3.08·53-s − 22.8·55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 1.71·5-s − 0.377·7-s + 2.06·9-s + 1.79·11-s + 0.400·13-s + 2.99·15-s + 1.47·17-s + 0.0170·19-s + 0.661·21-s + 0.924·23-s + 1.92·25-s − 1.85·27-s − 0.357·29-s − 0.316·31-s − 3.14·33-s + 0.646·35-s + 0.0817·37-s − 0.701·39-s − 0.156·41-s − 0.625·43-s − 3.53·45-s + 0.426·47-s + 0.142·49-s − 2.57·51-s + 0.423·53-s − 3.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7772238208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7772238208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 - 0.0743T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 - 0.497T + 37T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 5.87T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 0.670T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.139353239804312349973907252949, −7.36356426185801240607380000251, −6.80350666913074179728486576001, −6.24360951241040493022299407780, −5.37611600547111637047204317991, −4.62089140546959245504705686945, −3.81609951633976338602246395108, −3.41906548470000964824115339687, −1.30505209810060908901853831638, −0.63061898770631513457557339606,
0.63061898770631513457557339606, 1.30505209810060908901853831638, 3.41906548470000964824115339687, 3.81609951633976338602246395108, 4.62089140546959245504705686945, 5.37611600547111637047204317991, 6.24360951241040493022299407780, 6.80350666913074179728486576001, 7.36356426185801240607380000251, 8.139353239804312349973907252949
