| L(s) = 1 | + 0.494·3-s − 1.23·5-s − 1.36·7-s − 2.75·9-s + 4.69·11-s − 0.0431·13-s − 0.610·15-s − 7.31·17-s + 0.697·19-s − 0.676·21-s − 1.41·23-s − 3.47·25-s − 2.84·27-s + 8.30·29-s − 3.39·31-s + 2.32·33-s + 1.68·35-s + 7.15·37-s − 0.0213·39-s + 5.45·41-s + 11.7·43-s + 3.39·45-s + 5.24·47-s − 5.13·49-s − 3.61·51-s − 8.57·53-s − 5.79·55-s + ⋯ |
| L(s) = 1 | + 0.285·3-s − 0.551·5-s − 0.516·7-s − 0.918·9-s + 1.41·11-s − 0.0119·13-s − 0.157·15-s − 1.77·17-s + 0.160·19-s − 0.147·21-s − 0.294·23-s − 0.695·25-s − 0.548·27-s + 1.54·29-s − 0.610·31-s + 0.404·33-s + 0.284·35-s + 1.17·37-s − 0.00342·39-s + 0.851·41-s + 1.79·43-s + 0.506·45-s + 0.765·47-s − 0.733·49-s − 0.506·51-s − 1.17·53-s − 0.781·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427069958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427069958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 3 | \( 1 - 0.494T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 + 0.0431T + 13T^{2} \) |
| 17 | \( 1 + 7.31T + 17T^{2} \) |
| 19 | \( 1 - 0.697T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 - 5.45T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 2.32T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 2.23T + 97T^{2} \) |
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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
−8.504690146313602188665639906794, −7.88401517298461587811465482029, −6.89541967732588518449807333606, −6.37878305378134488365182381508, −5.65757010450176881999016714424, −4.35615286265816957019902022864, −3.99873845712334477634675238974, −2.96914141157201498780219728982, −2.15026280681035459998027339423, −0.66092063040697183094228513006,
0.66092063040697183094228513006, 2.15026280681035459998027339423, 2.96914141157201498780219728982, 3.99873845712334477634675238974, 4.35615286265816957019902022864, 5.65757010450176881999016714424, 6.37878305378134488365182381508, 6.89541967732588518449807333606, 7.88401517298461587811465482029, 8.504690146313602188665639906794
