| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s + 7·13-s + 15-s − 17-s + 3·19-s + 21-s + 9·23-s − 4·25-s − 27-s − 6·29-s − 4·31-s + 33-s + 35-s + 10·37-s − 7·39-s + 3·41-s + 3·43-s − 45-s + 8·47-s + 49-s + 51-s + 4·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.94·13-s + 0.258·15-s − 0.242·17-s + 0.688·19-s + 0.218·21-s + 1.87·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.169·35-s + 1.64·37-s − 1.12·39-s + 0.468·41-s + 0.457·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.549·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816950125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816950125\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.50945933312348, −15.14420849058128, −14.50274347781576, −13.62563306785478, −13.21706192319623, −12.95205663329067, −12.21682540270475, −11.43033039259815, −11.16443017667326, −10.83386278803026, −10.02852404151763, −9.325639492718607, −8.913175798633979, −8.242933603843858, −7.472479239145707, −7.116560411187217, −6.320892251960547, −5.721120722834323, −5.378361246177147, −4.344016552281744, −3.847122709586859, −3.251544845647157, −2.355927018535138, −1.243936751790859, −0.6444449955030485,
0.6444449955030485, 1.243936751790859, 2.355927018535138, 3.251544845647157, 3.847122709586859, 4.344016552281744, 5.378361246177147, 5.721120722834323, 6.320892251960547, 7.116560411187217, 7.472479239145707, 8.242933603843858, 8.913175798633979, 9.325639492718607, 10.02852404151763, 10.83386278803026, 11.16443017667326, 11.43033039259815, 12.21682540270475, 12.95205663329067, 13.21706192319623, 13.62563306785478, 14.50274347781576, 15.14420849058128, 15.50945933312348
