| L(s) = 1 | + 2-s − 2·3-s + 4-s + 1.73·5-s − 2·6-s − 3.46·7-s + 8-s + 9-s + 1.73·10-s + 3.46·11-s − 2·12-s − 5·13-s − 3.46·14-s − 3.46·15-s + 16-s − 6.92·17-s + 18-s + 3.46·19-s + 1.73·20-s + 6.92·21-s + 3.46·22-s − 2·24-s − 2.00·25-s − 5·26-s + 4·27-s − 3.46·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.774·5-s − 0.816·6-s − 1.30·7-s + 0.353·8-s + 0.333·9-s + 0.547·10-s + 1.04·11-s − 0.577·12-s − 1.38·13-s − 0.925·14-s − 0.894·15-s + 0.250·16-s − 1.68·17-s + 0.235·18-s + 0.794·19-s + 0.387·20-s + 1.51·21-s + 0.738·22-s − 0.408·24-s − 0.400·25-s − 0.980·26-s + 0.769·27-s − 0.654·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.609683689994221262995820314725, −9.030423524274746879434261281310, −7.25791999773504074219324103891, −6.66929610068002864912761735271, −6.03872479858973991238498336879, −5.32116971072219090775121203838, −4.38040337421304322935817548705, −3.19090143678034863696201098610, −1.96431640128010856232497960694, 0,
1.96431640128010856232497960694, 3.19090143678034863696201098610, 4.38040337421304322935817548705, 5.32116971072219090775121203838, 6.03872479858973991238498336879, 6.66929610068002864912761735271, 7.25791999773504074219324103891, 9.030423524274746879434261281310, 9.609683689994221262995820314725
