| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s + 4·11-s + 13-s + 2·15-s + 2·17-s − 4·21-s + 6·23-s + 25-s − 4·27-s + 10·29-s + 8·33-s − 2·35-s − 10·37-s + 2·39-s − 2·41-s + 2·43-s + 45-s + 6·47-s − 3·49-s + 4·51-s − 2·53-s + 4·55-s − 8·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s + 1.39·33-s − 0.338·35-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.539·55-s − 1.04·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.297155222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.297155222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.557961638033010612024877894844, −7.83418018174343538770830592379, −6.80494192027376907897344418282, −6.48248464077599487995018736826, −5.48242479293515865313722391669, −4.51648198364736228588663559652, −3.44338871890022594879128174489, −3.14531231435833438293403663927, −2.08689685350756469672706577876, −1.02788500595544746813779230476,
1.02788500595544746813779230476, 2.08689685350756469672706577876, 3.14531231435833438293403663927, 3.44338871890022594879128174489, 4.51648198364736228588663559652, 5.48242479293515865313722391669, 6.48248464077599487995018736826, 6.80494192027376907897344418282, 7.83418018174343538770830592379, 8.557961638033010612024877894844
