| L(s) = 1 | + 2.03·2-s − 3-s + 2.12·4-s + 5-s − 2.03·6-s − 0.0759·7-s + 0.251·8-s + 9-s + 2.03·10-s + 0.961·11-s − 2.12·12-s − 2.83·13-s − 0.154·14-s − 15-s − 3.73·16-s − 3.10·17-s + 2.03·18-s − 2.23·19-s + 2.12·20-s + 0.0759·21-s + 1.95·22-s + 5.88·23-s − 0.251·24-s + 25-s − 5.76·26-s − 27-s − 0.161·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.577·3-s + 1.06·4-s + 0.447·5-s − 0.829·6-s − 0.0286·7-s + 0.0888·8-s + 0.333·9-s + 0.642·10-s + 0.289·11-s − 0.613·12-s − 0.786·13-s − 0.0411·14-s − 0.258·15-s − 0.934·16-s − 0.753·17-s + 0.478·18-s − 0.513·19-s + 0.474·20-s + 0.0165·21-s + 0.416·22-s + 1.22·23-s − 0.0512·24-s + 0.200·25-s − 1.13·26-s − 0.192·27-s − 0.0304·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 7 | \( 1 + 0.0759T + 7T^{2} \) |
| 11 | \( 1 - 0.961T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 5.88T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 + 6.56T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 0.556T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 67 | \( 1 - 1.05T + 67T^{2} \) |
| 71 | \( 1 - 9.44T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 7.24T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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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
−7.19979789078601894654378234596, −6.79208937879093277634748149049, −6.19041837378580703605221521207, −5.35026322650172111416551511545, −4.92120622621466804165693497409, −4.27675324461431893182792714152, −3.36652635466603955778845646654, −2.56026767981170842649349028932, −1.62063551436664312191682737915, 0,
1.62063551436664312191682737915, 2.56026767981170842649349028932, 3.36652635466603955778845646654, 4.27675324461431893182792714152, 4.92120622621466804165693497409, 5.35026322650172111416551511545, 6.19041837378580703605221521207, 6.79208937879093277634748149049, 7.19979789078601894654378234596
