| L(s) = 1 | + 3-s − 0.585·5-s − 4.82·7-s + 9-s + 2.82·11-s − 4.82·13-s − 0.585·15-s − 3.41·17-s + 5.65·19-s − 4.82·21-s − 5.41·23-s − 4.65·25-s + 27-s − 1.75·29-s − 6.82·31-s + 2.82·33-s + 2.82·35-s − 37-s − 4.82·39-s − 11.6·41-s − 2.82·43-s − 0.585·45-s + 5.65·47-s + 16.3·49-s − 3.41·51-s − 12.8·53-s − 1.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.261·5-s − 1.82·7-s + 0.333·9-s + 0.852·11-s − 1.33·13-s − 0.151·15-s − 0.828·17-s + 1.29·19-s − 1.05·21-s − 1.12·23-s − 0.931·25-s + 0.192·27-s − 0.326·29-s − 1.22·31-s + 0.492·33-s + 0.478·35-s − 0.164·37-s − 0.773·39-s − 1.82·41-s − 0.431·43-s − 0.0873·45-s + 0.825·47-s + 2.33·49-s − 0.478·51-s − 1.76·53-s − 0.223·55-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 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 0.485T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.579507885627775546323840448126, −9.195680100069341484459232340047, −7.962199603608401979576509164204, −7.06922861097897143725260519758, −6.49361818465513172144524720539, −5.31790321228225452677182245349, −3.93498839307269095925174637778, −3.33144520256563759123046508682, −2.13293635491393479106156780994, 0,
2.13293635491393479106156780994, 3.33144520256563759123046508682, 3.93498839307269095925174637778, 5.31790321228225452677182245349, 6.49361818465513172144524720539, 7.06922861097897143725260519758, 7.962199603608401979576509164204, 9.195680100069341484459232340047, 9.579507885627775546323840448126
