| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 6·13-s − 15-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 27-s + 2·29-s + 31-s + 4·33-s − 2·37-s + 6·39-s − 6·41-s + 4·43-s − 45-s − 7·49-s + 2·51-s + 2·53-s − 4·55-s + 4·57-s + 4·59-s − 6·61-s − 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.179·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.280·51-s + 0.274·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.174659720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.174659720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.058059502099382169391027617641, −7.17570310625688789070085210236, −6.62109742024884721185378041953, −5.90206031249304209874587775267, −4.99792423429612852172868870321, −4.10496226597783123965370489820, −3.51766256594621741473528624836, −2.97604273249890037894404030298, −1.58356541259928707242394937292, −0.984584146938265637714044213661,
0.984584146938265637714044213661, 1.58356541259928707242394937292, 2.97604273249890037894404030298, 3.51766256594621741473528624836, 4.10496226597783123965370489820, 4.99792423429612852172868870321, 5.90206031249304209874587775267, 6.62109742024884721185378041953, 7.17570310625688789070085210236, 8.058059502099382169391027617641
