| L(s) = 1 | − 2.04·3-s − 4.02·5-s − 1.43·7-s + 1.16·9-s + 4.74·11-s + 2.52·13-s + 8.22·15-s + 2.51·17-s − 3.36·19-s + 2.92·21-s − 2.10·23-s + 11.2·25-s + 3.73·27-s + 3.73·29-s + 9.98·31-s − 9.68·33-s + 5.77·35-s + 11.0·37-s − 5.15·39-s − 11.3·41-s + 2.86·43-s − 4.70·45-s − 5.89·47-s − 4.94·49-s − 5.12·51-s − 10.3·53-s − 19.1·55-s + ⋯ |
| L(s) = 1 | − 1.17·3-s − 1.80·5-s − 0.541·7-s + 0.389·9-s + 1.43·11-s + 0.700·13-s + 2.12·15-s + 0.609·17-s − 0.772·19-s + 0.638·21-s − 0.439·23-s + 2.24·25-s + 0.719·27-s + 0.693·29-s + 1.79·31-s − 1.68·33-s + 0.975·35-s + 1.81·37-s − 0.825·39-s − 1.77·41-s + 0.437·43-s − 0.701·45-s − 0.859·47-s − 0.706·49-s − 0.718·51-s − 1.42·53-s − 2.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6890585104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6890585104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 + 3.36T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 9.98T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 0.398T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.60T + 89T^{2} \) |
| 97 | \( 1 + 3.96T + 97T^{2} \) |
| 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
−8.039319518950491489503425906100, −6.86117289656225919254912185570, −6.42212647274108595704644956521, −6.06647325555670359906448639277, −4.73592437894407473841992342890, −4.45386491268939290765611382264, −3.59116554601977020910235247883, −3.03245829672370194927639195317, −1.29454376475132915345281165078, −0.49670924964192781792333849778,
0.49670924964192781792333849778, 1.29454376475132915345281165078, 3.03245829672370194927639195317, 3.59116554601977020910235247883, 4.45386491268939290765611382264, 4.73592437894407473841992342890, 6.06647325555670359906448639277, 6.42212647274108595704644956521, 6.86117289656225919254912185570, 8.039319518950491489503425906100
