| L(s) = 1 | − 1.67·3-s + 5-s − 7-s − 0.193·9-s − 11-s + 1.67·13-s − 1.67·15-s + 0.324·17-s − 3.61·19-s + 1.67·21-s − 0.806·23-s + 25-s + 5.35·27-s + 7.92·29-s − 1.13·31-s + 1.67·33-s − 35-s − 7.76·37-s − 2.80·39-s + 8.21·41-s − 5.11·43-s − 0.193·45-s + 2.24·47-s + 49-s − 0.544·51-s + 10.4·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.967·3-s + 0.447·5-s − 0.377·7-s − 0.0646·9-s − 0.301·11-s + 0.464·13-s − 0.432·15-s + 0.0787·17-s − 0.828·19-s + 0.365·21-s − 0.168·23-s + 0.200·25-s + 1.02·27-s + 1.47·29-s − 0.203·31-s + 0.291·33-s − 0.169·35-s − 1.27·37-s − 0.449·39-s + 1.28·41-s − 0.780·43-s − 0.0289·45-s + 0.328·47-s + 0.142·49-s − 0.0762·51-s + 1.43·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 - 0.324T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 0.806T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 7.76T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 + 5.11T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 - 0.775T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 6.05T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−7.62760047030806862949738300578, −6.69582522172933023279922558728, −6.28315703552141406568054058349, −5.64298841901723369976279906887, −4.98437985279056674559627387436, −4.18290505269010634935295867516, −3.16071414513566334705187683012, −2.31920243348172436839863965958, −1.13151240244336754190049472111, 0,
1.13151240244336754190049472111, 2.31920243348172436839863965958, 3.16071414513566334705187683012, 4.18290505269010634935295867516, 4.98437985279056674559627387436, 5.64298841901723369976279906887, 6.28315703552141406568054058349, 6.69582522172933023279922558728, 7.62760047030806862949738300578
