| L(s) = 1 | + 3-s − 3.41·5-s + 0.828·7-s + 9-s − 2.82·11-s + 0.828·13-s − 3.41·15-s − 0.585·17-s − 5.65·19-s + 0.828·21-s − 2.58·23-s + 6.65·25-s + 27-s − 10.2·29-s − 1.17·31-s − 2.82·33-s − 2.82·35-s − 37-s + 0.828·39-s − 0.343·41-s + 2.82·43-s − 3.41·45-s − 5.65·47-s − 6.31·49-s − 0.585·51-s − 7.17·53-s + 9.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.52·5-s + 0.313·7-s + 0.333·9-s − 0.852·11-s + 0.229·13-s − 0.881·15-s − 0.142·17-s − 1.29·19-s + 0.180·21-s − 0.539·23-s + 1.33·25-s + 0.192·27-s − 1.90·29-s − 0.210·31-s − 0.492·33-s − 0.478·35-s − 0.164·37-s + 0.132·39-s − 0.0535·41-s + 0.431·43-s − 0.508·45-s − 0.825·47-s − 0.901·49-s − 0.0820·51-s − 0.985·53-s + 1.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 7.17T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 0.828T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−9.605708302810561857543857926625, −8.631773565262735651496500130349, −7.958712784038577791958502206703, −7.52653331330372259265694680402, −6.39873199918971056593385782247, −5.04871420234573366796349749819, −4.11840636468448025348553360054, −3.38288832110874083387379337877, −2.04585992883107583908744567298, 0,
2.04585992883107583908744567298, 3.38288832110874083387379337877, 4.11840636468448025348553360054, 5.04871420234573366796349749819, 6.39873199918971056593385782247, 7.52653331330372259265694680402, 7.958712784038577791958502206703, 8.631773565262735651496500130349, 9.605708302810561857543857926625
