| L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 9-s − 4·11-s + 4·13-s − 2·15-s − 2·17-s + 8·21-s + 25-s + 4·27-s − 2·29-s + 8·31-s + 8·33-s − 4·35-s − 4·37-s − 8·39-s − 6·41-s + 45-s − 12·47-s + 9·49-s + 4·51-s + 8·53-s − 4·55-s + 2·61-s − 4·63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.516·15-s − 0.485·17-s + 1.74·21-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 1.43·31-s + 1.39·33-s − 0.676·35-s − 0.657·37-s − 1.28·39-s − 0.937·41-s + 0.149·45-s − 1.75·47-s + 9/7·49-s + 0.560·51-s + 1.09·53-s − 0.539·55-s + 0.256·61-s − 0.503·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.49811377060672, −15.32262132216010, −14.22607252881417, −13.64509275335093, −13.16686697676877, −12.93405339030210, −12.27827620745365, −11.64950951331603, −11.19283170388923, −10.48354316472272, −10.20390583997847, −9.752672319804356, −8.865647645517930, −8.503675090696058, −7.687395250940399, −6.779906651191811, −6.510906310798410, −6.015451768846244, −5.440819386299738, −4.937471883930857, −4.096782127940903, −3.217880204377849, −2.815257313505841, −1.803626499120790, −0.7266882887456315, 0,
0.7266882887456315, 1.803626499120790, 2.815257313505841, 3.217880204377849, 4.096782127940903, 4.937471883930857, 5.440819386299738, 6.015451768846244, 6.510906310798410, 6.779906651191811, 7.687395250940399, 8.503675090696058, 8.865647645517930, 9.752672319804356, 10.20390583997847, 10.48354316472272, 11.19283170388923, 11.64950951331603, 12.27827620745365, 12.93405339030210, 13.16686697676877, 13.64509275335093, 14.22607252881417, 15.32262132216010, 15.49811377060672
