| L(s) = 1 | − 1.83·2-s + 1.35·4-s + 2.62·5-s + 1.17·8-s − 4.81·10-s − 3.26·11-s − 13-s − 4.87·16-s + 4.53·17-s − 4.06·19-s + 3.56·20-s + 5.98·22-s + 4.53·23-s + 1.89·25-s + 1.83·26-s + 1.42·29-s + 2.80·31-s + 6.57·32-s − 8.30·34-s − 10.0·37-s + 7.45·38-s + 3.09·40-s − 2.84·41-s + 9.72·43-s − 4.43·44-s − 8.30·46-s − 9.44·47-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.678·4-s + 1.17·5-s + 0.416·8-s − 1.52·10-s − 0.984·11-s − 0.277·13-s − 1.21·16-s + 1.09·17-s − 0.932·19-s + 0.797·20-s + 1.27·22-s + 0.945·23-s + 0.378·25-s + 0.359·26-s + 0.264·29-s + 0.503·31-s + 1.16·32-s − 1.42·34-s − 1.65·37-s + 1.20·38-s + 0.488·40-s − 0.443·41-s + 1.48·43-s − 0.668·44-s − 1.22·46-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039115896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039115896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 - 1.42T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 - 9.72T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 + 5.26T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.98T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.189518092092337796390604676801, −7.64069444505446334987892182533, −6.84708458594494153019180149359, −6.15092195925975388522736674016, −5.23749784812395022983447379242, −4.76817387816455889574631264245, −3.40164856458727653680568669595, −2.39119999172243971889188972500, −1.74545910857162190758963644079, −0.66476768298859719190159150557,
0.66476768298859719190159150557, 1.74545910857162190758963644079, 2.39119999172243971889188972500, 3.40164856458727653680568669595, 4.76817387816455889574631264245, 5.23749784812395022983447379242, 6.15092195925975388522736674016, 6.84708458594494153019180149359, 7.64069444505446334987892182533, 8.189518092092337796390604676801
