| L(s) = 1 | + 3-s − 0.585·5-s − 7-s + 9-s + 2.82·11-s − 5.65·13-s − 0.585·15-s − 6.24·17-s + 19-s − 21-s + 8.48·23-s − 4.65·25-s + 27-s + 1.75·29-s + 3.17·31-s + 2.82·33-s + 0.585·35-s − 6·37-s − 5.65·39-s − 3.17·41-s − 3.17·43-s − 0.585·45-s − 13.4·47-s + 49-s − 6.24·51-s + 9.07·53-s − 1.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.261·5-s − 0.377·7-s + 0.333·9-s + 0.852·11-s − 1.56·13-s − 0.151·15-s − 1.51·17-s + 0.229·19-s − 0.218·21-s + 1.76·23-s − 0.931·25-s + 0.192·27-s + 0.326·29-s + 0.569·31-s + 0.492·33-s + 0.0990·35-s − 0.986·37-s − 0.905·39-s − 0.495·41-s − 0.483·43-s − 0.0873·45-s − 1.95·47-s + 0.142·49-s − 0.874·51-s + 1.24·53-s − 0.223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 3.75T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.462399436032068907590945347015, −7.37413780590136710789175356611, −6.98997765969264461242392868366, −6.23143646501705730214391949789, −4.94701896220361361764655710890, −4.47071256086318044548912069066, −3.40258510178812708964369422359, −2.66814985854442093997786777427, −1.61911530681874039405030078714, 0,
1.61911530681874039405030078714, 2.66814985854442093997786777427, 3.40258510178812708964369422359, 4.47071256086318044548912069066, 4.94701896220361361764655710890, 6.23143646501705730214391949789, 6.98997765969264461242392868366, 7.37413780590136710789175356611, 8.462399436032068907590945347015
