| L(s) = 1 | − 1.71·3-s + 3.42·7-s − 0.0574·9-s + 5.34·11-s + 3.52·13-s − 2.55·17-s + 2.02·19-s − 5.87·21-s + 7.57·23-s + 5.24·27-s + 4.74·29-s − 1.62·31-s − 9.16·33-s + 0.0134·37-s − 6.04·39-s + 9.67·41-s + 2.32·43-s + 6.94·47-s + 4.72·49-s + 4.38·51-s − 1.72·53-s − 3.47·57-s + 0.0221·59-s + 3.91·61-s − 0.196·63-s + 4.11·67-s − 12.9·69-s + ⋯ |
| L(s) = 1 | − 0.990·3-s + 1.29·7-s − 0.0191·9-s + 1.61·11-s + 0.976·13-s − 0.620·17-s + 0.464·19-s − 1.28·21-s + 1.57·23-s + 1.00·27-s + 0.880·29-s − 0.291·31-s − 1.59·33-s + 0.00220·37-s − 0.967·39-s + 1.51·41-s + 0.354·43-s + 1.01·47-s + 0.674·49-s + 0.614·51-s − 0.236·53-s − 0.460·57-s + 0.00288·59-s + 0.501·61-s − 0.0247·63-s + 0.503·67-s − 1.56·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328702981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328702981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 + 1.62T + 31T^{2} \) |
| 37 | \( 1 - 0.0134T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 - 0.0221T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 1.51T + 73T^{2} \) |
| 79 | \( 1 + 0.426T + 79T^{2} \) |
| 83 | \( 1 + 6.04T + 83T^{2} \) |
| 89 | \( 1 + 6.09T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
| 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
−7.51737160296745600756043691548, −6.85241715941579921336730500112, −6.24338462177052022906083862356, −5.66496601261990028546681795973, −4.89151366647674848538471716283, −4.38219328516378505301461646493, −3.58906505583640500024010765535, −2.50772450429575725113993959931, −1.30515460554300633249344669054, −0.927959329550112401093090746079,
0.927959329550112401093090746079, 1.30515460554300633249344669054, 2.50772450429575725113993959931, 3.58906505583640500024010765535, 4.38219328516378505301461646493, 4.89151366647674848538471716283, 5.66496601261990028546681795973, 6.24338462177052022906083862356, 6.85241715941579921336730500112, 7.51737160296745600756043691548
