| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 4.82·7-s − 1.58·8-s + 11-s − 5.65·13-s + 1.99·14-s + 3·16-s − 6.82·17-s − 1.17·19-s + 0.414·22-s − 4·23-s − 2.34·26-s − 8.82·28-s − 0.828·29-s + 4.41·32-s − 2.82·34-s − 0.343·37-s − 0.485·38-s + 0.828·41-s + 3.17·43-s − 1.82·44-s − 1.65·46-s − 4·47-s + 16.3·49-s + 10.3·52-s − 13.3·53-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 1.82·7-s − 0.560·8-s + 0.301·11-s − 1.56·13-s + 0.534·14-s + 0.750·16-s − 1.65·17-s − 0.268·19-s + 0.0883·22-s − 0.834·23-s − 0.459·26-s − 1.66·28-s − 0.153·29-s + 0.780·32-s − 0.485·34-s − 0.0564·37-s − 0.0787·38-s + 0.129·41-s + 0.483·43-s − 0.275·44-s − 0.244·46-s − 0.583·47-s + 2.33·49-s + 1.43·52-s − 1.82·53-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 0.343T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−8.513919456537368504398299507875, −7.916052298769149406857064929715, −7.14448861378128696754356355404, −6.04352277637619096256287229548, −5.07494275294566641492761933675, −4.61189264607412807046133003673, −4.07211395306853007804500071477, −2.57677189011361346570474515760, −1.64464510029904934324792656499, 0,
1.64464510029904934324792656499, 2.57677189011361346570474515760, 4.07211395306853007804500071477, 4.61189264607412807046133003673, 5.07494275294566641492761933675, 6.04352277637619096256287229548, 7.14448861378128696754356355404, 7.916052298769149406857064929715, 8.513919456537368504398299507875
