| L(s) = 1 | + 1.29·2-s + 0.350·3-s − 0.335·4-s − 5-s + 0.452·6-s − 3.01·8-s − 2.87·9-s − 1.29·10-s + 11-s − 0.117·12-s + 3.47·13-s − 0.350·15-s − 3.21·16-s + 2.43·17-s − 3.71·18-s − 4.88·19-s + 0.335·20-s + 1.29·22-s + 6.56·23-s − 1.05·24-s + 25-s + 4.47·26-s − 2.06·27-s + 6.89·29-s − 0.452·30-s − 5.65·31-s + 1.87·32-s + ⋯ |
| L(s) = 1 | + 0.912·2-s + 0.202·3-s − 0.167·4-s − 0.447·5-s + 0.184·6-s − 1.06·8-s − 0.958·9-s − 0.407·10-s + 0.301·11-s − 0.0340·12-s + 0.962·13-s − 0.0905·15-s − 0.803·16-s + 0.591·17-s − 0.874·18-s − 1.12·19-s + 0.0750·20-s + 0.275·22-s + 1.36·23-s − 0.215·24-s + 0.200·25-s + 0.878·26-s − 0.396·27-s + 1.28·29-s − 0.0826·30-s − 1.01·31-s + 0.332·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.222004332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.222004332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 3 | \( 1 - 0.350T + 3T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 8.05T + 37T^{2} \) |
| 41 | \( 1 - 5.48T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 - 1.89T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + 5.51T + 89T^{2} \) |
| 97 | \( 1 - 9.66T + 97T^{2} \) |
| 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
−8.750898233621650469301673567756, −8.290526519550981430908343573560, −7.23654568882659093121576805962, −6.22774518774266215279148770246, −5.76961511576320289318697983552, −4.80422586759637508758771684296, −4.05084232502270504713051589715, −3.32035347877031053089248424893, −2.54967392271844509992946684953, −0.805264118038250007959944193226,
0.805264118038250007959944193226, 2.54967392271844509992946684953, 3.32035347877031053089248424893, 4.05084232502270504713051589715, 4.80422586759637508758771684296, 5.76961511576320289318697983552, 6.22774518774266215279148770246, 7.23654568882659093121576805962, 8.290526519550981430908343573560, 8.750898233621650469301673567756
