| L(s) = 1 | + 3-s + 4·5-s + 4·7-s + 9-s − 2·11-s + 13-s + 4·15-s − 6·17-s + 4·19-s + 4·21-s − 4·23-s + 11·25-s + 27-s + 6·29-s − 8·31-s − 2·33-s + 16·35-s + 10·37-s + 39-s − 4·41-s − 4·43-s + 4·45-s + 6·47-s + 9·49-s − 6·51-s − 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 1.03·15-s − 1.45·17-s + 0.917·19-s + 0.872·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.348·33-s + 2.70·35-s + 1.64·37-s + 0.160·39-s − 0.624·41-s − 0.609·43-s + 0.596·45-s + 0.875·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.657224129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.657224129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.929245164931267916957501525813, −8.280639349950524940517545786655, −7.50697019610482934951543843065, −6.55745468953340788447724539331, −5.71945105008069028214353159467, −5.03891234227654532641353465088, −4.29907608999458989347549727155, −2.81340920444160493801929129628, −2.07708294910507683638034694156, −1.39846104434260481514027206509,
1.39846104434260481514027206509, 2.07708294910507683638034694156, 2.81340920444160493801929129628, 4.29907608999458989347549727155, 5.03891234227654532641353465088, 5.71945105008069028214353159467, 6.55745468953340788447724539331, 7.50697019610482934951543843065, 8.280639349950524940517545786655, 8.929245164931267916957501525813
