| L(s) = 1 | + 3-s − 3·5-s + 7-s − 2·9-s + 11-s − 13-s − 3·15-s + 21-s + 7·23-s + 4·25-s − 5·27-s + 7·31-s + 33-s − 3·35-s − 7·37-s − 39-s − 4·41-s + 2·43-s + 6·45-s + 49-s − 10·53-s − 3·55-s + 9·59-s − 12·61-s − 2·63-s + 3·65-s + 7·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 0.774·15-s + 0.218·21-s + 1.45·23-s + 4/5·25-s − 0.962·27-s + 1.25·31-s + 0.174·33-s − 0.507·35-s − 1.15·37-s − 0.160·39-s − 0.624·41-s + 0.304·43-s + 0.894·45-s + 1/7·49-s − 1.37·53-s − 0.404·55-s + 1.17·59-s − 1.53·61-s − 0.251·63-s + 0.372·65-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−16.09634645415147, −15.46259682612092, −15.22642592709106, −14.65287247420887, −14.07613967999595, −13.65503131010869, −12.83339582031693, −12.23221103103273, −11.77780932937950, −11.20536881106205, −10.85518768784960, −9.961340025242871, −9.267738269090244, −8.641004483144630, −8.270308559168812, −7.720240021359553, −7.093468317639440, −6.517270233719568, −5.545532682456316, −4.850580684774637, −4.289654655685183, −3.415813577976323, −3.079557384957085, −2.148014174500918, −1.048161784731548, 0,
1.048161784731548, 2.148014174500918, 3.079557384957085, 3.415813577976323, 4.289654655685183, 4.850580684774637, 5.545532682456316, 6.517270233719568, 7.093468317639440, 7.720240021359553, 8.270308559168812, 8.641004483144630, 9.267738269090244, 9.961340025242871, 10.85518768784960, 11.20536881106205, 11.77780932937950, 12.23221103103273, 12.83339582031693, 13.65503131010869, 14.07613967999595, 14.65287247420887, 15.22642592709106, 15.46259682612092, 16.09634645415147
