| L(s) = 1 | − 0.968·2-s − 1.06·4-s + 2.59·5-s + 3.83·7-s + 2.96·8-s − 2.51·10-s + 0.279·11-s − 4.21·13-s − 3.71·14-s − 0.748·16-s + 7.81·17-s + 5.87·19-s − 2.75·20-s − 0.270·22-s + 3.65·23-s + 1.74·25-s + 4.08·26-s − 4.07·28-s + 0.344·29-s + 6.17·31-s − 5.20·32-s − 7.56·34-s + 9.95·35-s − 3.88·37-s − 5.69·38-s + 7.70·40-s + 2.01·41-s + ⋯ |
| L(s) = 1 | − 0.684·2-s − 0.530·4-s + 1.16·5-s + 1.44·7-s + 1.04·8-s − 0.795·10-s + 0.0842·11-s − 1.16·13-s − 0.992·14-s − 0.187·16-s + 1.89·17-s + 1.34·19-s − 0.616·20-s − 0.0577·22-s + 0.761·23-s + 0.348·25-s + 0.801·26-s − 0.769·28-s + 0.0639·29-s + 1.10·31-s − 0.920·32-s − 1.29·34-s + 1.68·35-s − 0.639·37-s − 0.923·38-s + 1.21·40-s + 0.314·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.101107321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101107321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 + 0.968T + 2T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 0.279T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 - 7.81T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 0.344T + 29T^{2} \) |
| 31 | \( 1 - 6.17T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 3.50T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 3.28T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.117036334924288724150348758492, −7.57823025336777583704276863147, −6.92920184465090559461712319962, −5.66246939277490784238859948914, −5.12951826212970437424860981248, −4.85369002554474232821992976845, −3.61045543316978156685614997225, −2.52424477156339156470679133457, −1.52906580414256804472955178512, −0.977465380623172982007555300489,
0.977465380623172982007555300489, 1.52906580414256804472955178512, 2.52424477156339156470679133457, 3.61045543316978156685614997225, 4.85369002554474232821992976845, 5.12951826212970437424860981248, 5.66246939277490784238859948914, 6.92920184465090559461712319962, 7.57823025336777583704276863147, 8.117036334924288724150348758492
