| L(s) = 1 | − 1.15·2-s − 3-s − 0.662·4-s − 1.96·5-s + 1.15·6-s + 2.38·7-s + 3.07·8-s + 9-s + 2.26·10-s + 11-s + 0.662·12-s − 2.75·14-s + 1.96·15-s − 2.23·16-s + 7.24·17-s − 1.15·18-s − 3.58·19-s + 1.29·20-s − 2.38·21-s − 1.15·22-s − 8.67·23-s − 3.07·24-s − 1.15·25-s − 27-s − 1.57·28-s + 5.68·29-s − 2.26·30-s + ⋯ |
| L(s) = 1 | − 0.817·2-s − 0.577·3-s − 0.331·4-s − 0.877·5-s + 0.472·6-s + 0.900·7-s + 1.08·8-s + 0.333·9-s + 0.717·10-s + 0.301·11-s + 0.191·12-s − 0.736·14-s + 0.506·15-s − 0.558·16-s + 1.75·17-s − 0.272·18-s − 0.821·19-s + 0.290·20-s − 0.520·21-s − 0.246·22-s − 1.80·23-s − 0.628·24-s − 0.230·25-s − 0.192·27-s − 0.298·28-s + 1.05·29-s − 0.414·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 17 | \( 1 - 7.24T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.24T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 0.393T + 67T^{2} \) |
| 71 | \( 1 + 5.42T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.85694116421955511486780125024, −7.52869438787398377189350809737, −6.38919161585236626881313221639, −5.64827026948158450960128077015, −4.71648482686949638676616934268, −4.25602541880528198636198605621, −3.45749748543313644698614639013, −1.92071260428994253980538606045, −1.07014722193450605850834493744, 0,
1.07014722193450605850834493744, 1.92071260428994253980538606045, 3.45749748543313644698614639013, 4.25602541880528198636198605621, 4.71648482686949638676616934268, 5.64827026948158450960128077015, 6.38919161585236626881313221639, 7.52869438787398377189350809737, 7.85694116421955511486780125024
