| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 2·7-s − 3·8-s + 9-s + 2·12-s + 13-s − 2·14-s − 16-s − 5·17-s + 18-s − 6·19-s + 4·21-s − 2·23-s + 6·24-s + 26-s + 4·27-s + 2·28-s − 9·29-s − 2·31-s + 5·32-s − 5·34-s − 36-s + 3·37-s − 6·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 1/4·16-s − 1.21·17-s + 0.235·18-s − 1.37·19-s + 0.872·21-s − 0.417·23-s + 1.22·24-s + 0.196·26-s + 0.769·27-s + 0.377·28-s − 1.67·29-s − 0.359·31-s + 0.883·32-s − 0.857·34-s − 1/6·36-s + 0.493·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4560661268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4560661268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.891544694938778377076997987391, −7.967007918919657438446010948975, −6.74595017018474395321939782199, −6.24003407315348404281547464047, −5.73923271721281018399677925756, −4.84771137662795168915150076413, −4.19484040485853443780535692496, −3.38984207611525607636997351029, −2.17355028292033769488167988726, −0.37074583497620754696034055659,
0.37074583497620754696034055659, 2.17355028292033769488167988726, 3.38984207611525607636997351029, 4.19484040485853443780535692496, 4.84771137662795168915150076413, 5.73923271721281018399677925756, 6.24003407315348404281547464047, 6.74595017018474395321939782199, 7.967007918919657438446010948975, 8.891544694938778377076997987391
