| L(s) = 1 | − 3-s + 4·5-s + 7-s + 9-s + 2·11-s + 2·13-s − 4·15-s + 4·19-s − 21-s − 6·23-s + 11·25-s − 27-s + 10·29-s − 8·31-s − 2·33-s + 4·35-s − 10·37-s − 2·39-s − 4·41-s + 8·43-s + 4·45-s − 4·47-s + 49-s − 10·53-s + 8·55-s − 4·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.03·15-s + 0.917·19-s − 0.218·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.348·33-s + 0.676·35-s − 1.64·37-s − 0.320·39-s − 0.624·41-s + 1.21·43-s + 0.596·45-s − 0.583·47-s + 1/7·49-s − 1.37·53-s + 1.07·55-s − 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162886616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162886616\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.630930440923585418449522891985, −9.052254644927438516803566226587, −8.052896223091606570950311686774, −6.82996158982273698048374301846, −6.24984441387476424451277097244, −5.51173614948836766111548042093, −4.82087120360185900706726704905, −3.48798700664607672936542140881, −2.08673280409477301775619832385, −1.25803993641506171158754813946,
1.25803993641506171158754813946, 2.08673280409477301775619832385, 3.48798700664607672936542140881, 4.82087120360185900706726704905, 5.51173614948836766111548042093, 6.24984441387476424451277097244, 6.82996158982273698048374301846, 8.052896223091606570950311686774, 9.052254644927438516803566226587, 9.630930440923585418449522891985
