| L(s) = 1 | + 1.55·2-s + 3-s + 0.420·4-s − 3.26·5-s + 1.55·6-s − 2.45·8-s + 9-s − 5.07·10-s − 5.43·11-s + 0.420·12-s − 5.23·13-s − 3.26·15-s − 4.66·16-s − 0.709·17-s + 1.55·18-s + 2.66·19-s − 1.37·20-s − 8.45·22-s + 5.52·23-s − 2.45·24-s + 5.65·25-s − 8.15·26-s + 27-s + 4.20·29-s − 5.07·30-s + 2.85·31-s − 2.34·32-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.210·4-s − 1.45·5-s + 0.635·6-s − 0.868·8-s + 0.333·9-s − 1.60·10-s − 1.63·11-s + 0.121·12-s − 1.45·13-s − 0.842·15-s − 1.16·16-s − 0.172·17-s + 0.366·18-s + 0.612·19-s − 0.307·20-s − 1.80·22-s + 1.15·23-s − 0.501·24-s + 1.13·25-s − 1.59·26-s + 0.192·27-s + 0.781·29-s − 0.927·30-s + 0.512·31-s − 0.414·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650572283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650572283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 0.709T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 5.49T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−8.078148257704322202442474070575, −7.30301412710632059245936900172, −6.86993339715624906682404007873, −5.60511643235484690400062808223, −4.83592139567319359860243227685, −4.62810105181135215129837760447, −3.63808209931588040483666874386, −2.92982360892648121014155655927, −2.49899218884091392790966845723, −0.52184376373507037061603481761,
0.52184376373507037061603481761, 2.49899218884091392790966845723, 2.92982360892648121014155655927, 3.63808209931588040483666874386, 4.62810105181135215129837760447, 4.83592139567319359860243227685, 5.60511643235484690400062808223, 6.86993339715624906682404007873, 7.30301412710632059245936900172, 8.078148257704322202442474070575
