| L(s) = 1 | − 1.99·2-s + 1.39·3-s + 1.98·4-s − 2.55·5-s − 2.77·6-s − 4.72·7-s + 0.0277·8-s − 1.06·9-s + 5.09·10-s + 6.31·11-s + 2.76·12-s + 4.90·13-s + 9.44·14-s − 3.54·15-s − 4.02·16-s + 5.32·17-s + 2.12·18-s + 5.28·19-s − 5.06·20-s − 6.57·21-s − 12.6·22-s + 2.28·23-s + 0.0385·24-s + 1.50·25-s − 9.78·26-s − 5.65·27-s − 9.39·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.802·3-s + 0.993·4-s − 1.14·5-s − 1.13·6-s − 1.78·7-s + 0.00979·8-s − 0.355·9-s + 1.61·10-s + 1.90·11-s + 0.797·12-s + 1.35·13-s + 2.52·14-s − 0.915·15-s − 1.00·16-s + 1.29·17-s + 0.501·18-s + 1.21·19-s − 1.13·20-s − 1.43·21-s − 2.68·22-s + 0.476·23-s + 0.00786·24-s + 0.301·25-s − 1.91·26-s − 1.08·27-s − 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6440475400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6440475400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 - T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 - 6.31T + 11T^{2} \) |
| 13 | \( 1 - 4.90T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 0.490T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 + 0.735T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 - 0.220T + 53T^{2} \) |
| 59 | \( 1 - 8.86T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 - 0.475T + 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 79 | \( 1 - 4.32T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−11.48228220381674948657805183293, −10.17221790403051186509909531371, −9.326602821048810051405781856937, −8.899320531148769735910076249197, −7.988775644037646775289614898589, −7.07852145938240370370017691465, −6.15588489202518323399345259421, −3.75679526750219236600599784608, −3.28492273042760871245376486146, −0.982598843298159794717654686575,
0.982598843298159794717654686575, 3.28492273042760871245376486146, 3.75679526750219236600599784608, 6.15588489202518323399345259421, 7.07852145938240370370017691465, 7.988775644037646775289614898589, 8.899320531148769735910076249197, 9.326602821048810051405781856937, 10.17221790403051186509909531371, 11.48228220381674948657805183293
