| L(s) = 1 | − 2·4-s − 7-s + 11-s − 13-s + 4·16-s + 6·17-s − 7·19-s − 6·23-s + 2·28-s + 6·29-s − 7·31-s + 2·37-s + 6·41-s − 43-s − 2·44-s − 6·49-s + 2·52-s + 6·53-s + 5·61-s − 8·64-s + 5·67-s − 12·68-s + 12·71-s + 14·73-s + 14·76-s − 77-s − 4·79-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 0.301·11-s − 0.277·13-s + 16-s + 1.45·17-s − 1.60·19-s − 1.25·23-s + 0.377·28-s + 1.11·29-s − 1.25·31-s + 0.328·37-s + 0.937·41-s − 0.152·43-s − 0.301·44-s − 6/7·49-s + 0.277·52-s + 0.824·53-s + 0.640·61-s − 64-s + 0.610·67-s − 1.45·68-s + 1.42·71-s + 1.63·73-s + 1.60·76-s − 0.113·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.117963749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117963749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.899119495915130256624258150613, −8.240178529480385739684701459320, −7.59106525614709698283393492786, −6.48339477315215306115657995580, −5.80741859506924874487989702702, −4.93966973271202037240525713331, −4.06083776331402365524867624470, −3.42870909379007946401799132003, −2.11432709634023786835252102513, −0.67233943724094663765995022815,
0.67233943724094663765995022815, 2.11432709634023786835252102513, 3.42870909379007946401799132003, 4.06083776331402365524867624470, 4.93966973271202037240525713331, 5.80741859506924874487989702702, 6.48339477315215306115657995580, 7.59106525614709698283393492786, 8.240178529480385739684701459320, 8.899119495915130256624258150613
