| L(s) = 1 | + 3-s + 1.82·5-s + 0.862·7-s + 9-s + 11-s + 13-s + 1.82·15-s + 0.551·17-s + 0.300·19-s + 0.862·21-s + 7.71·23-s − 1.65·25-s + 27-s − 4.56·29-s − 0.391·31-s + 33-s + 1.57·35-s + 9.60·37-s + 39-s + 5.29·41-s + 11.7·43-s + 1.82·45-s − 6.50·47-s − 6.25·49-s + 0.551·51-s − 9.76·53-s + 1.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.817·5-s + 0.326·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.472·15-s + 0.133·17-s + 0.0690·19-s + 0.188·21-s + 1.60·23-s − 0.330·25-s + 0.192·27-s − 0.847·29-s − 0.0702·31-s + 0.174·33-s + 0.266·35-s + 1.57·37-s + 0.160·39-s + 0.827·41-s + 1.79·43-s + 0.272·45-s − 0.949·47-s − 0.893·49-s + 0.0772·51-s − 1.34·53-s + 0.246·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.532482064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.532482064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 - 0.862T + 7T^{2} \) |
| 17 | \( 1 - 0.551T + 17T^{2} \) |
| 19 | \( 1 - 0.300T + 19T^{2} \) |
| 23 | \( 1 - 7.71T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 + 0.391T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 6.50T + 47T^{2} \) |
| 53 | \( 1 + 9.76T + 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 + 8.53T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 - 0.795T + 73T^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 + 5.49T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.82391348596419459478227971082, −7.48421191885414531866865743446, −6.45232263660216577055565817176, −5.96753137440084031249613958387, −5.09252947622980778679508992214, −4.39099725110607790853493717177, −3.47904546285437507671587120762, −2.68631713333425508368937827088, −1.85117304742605595957822933251, −0.994888079861440283058567961533,
0.994888079861440283058567961533, 1.85117304742605595957822933251, 2.68631713333425508368937827088, 3.47904546285437507671587120762, 4.39099725110607790853493717177, 5.09252947622980778679508992214, 5.96753137440084031249613958387, 6.45232263660216577055565817176, 7.48421191885414531866865743446, 7.82391348596419459478227971082
