L(s) = 1 | + 2-s + 3-s + 4-s + 2.32·5-s + 6-s − 3.70·7-s + 8-s + 9-s + 2.32·10-s + 6.03·11-s + 12-s − 13-s − 3.70·14-s + 2.32·15-s + 16-s + 7.20·17-s + 18-s + 0.841·19-s + 2.32·20-s − 3.70·21-s + 6.03·22-s + 2.94·23-s + 24-s + 0.401·25-s − 26-s + 27-s − 3.70·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.03·5-s + 0.408·6-s − 1.40·7-s + 0.353·8-s + 0.333·9-s + 0.734·10-s + 1.81·11-s + 0.288·12-s − 0.277·13-s − 0.990·14-s + 0.600·15-s + 0.250·16-s + 1.74·17-s + 0.235·18-s + 0.192·19-s + 0.519·20-s − 0.809·21-s + 1.28·22-s + 0.614·23-s + 0.204·24-s + 0.0802·25-s − 0.196·26-s + 0.192·27-s − 0.700·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.380871409\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.380871409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 - 6.03T + 11T^{2} \) |
| 17 | \( 1 - 7.20T + 17T^{2} \) |
| 19 | \( 1 - 0.841T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 9.92T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 + 0.782T + 59T^{2} \) |
| 61 | \( 1 - 5.91T + 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 + 4.98T + 71T^{2} \) |
| 73 | \( 1 + 4.01T + 73T^{2} \) |
| 79 | \( 1 - 7.92T + 79T^{2} \) |
| 83 | \( 1 + 8.29T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 - 2.64T + 97T^{2} \) |
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
−7.65592564221529577871983866374, −6.88843344100552684388089096053, −6.37299047790283347835657686865, −5.90610971465165334772691875958, −5.08659889963209906588140163067, −4.10862127280498995952015366204, −3.34225677277717910182538059864, −3.00707609628571947553025656490, −1.87117904347517330228278087718, −1.08347156483314789424691467887,
1.08347156483314789424691467887, 1.87117904347517330228278087718, 3.00707609628571947553025656490, 3.34225677277717910182538059864, 4.10862127280498995952015366204, 5.08659889963209906588140163067, 5.90610971465165334772691875958, 6.37299047790283347835657686865, 6.88843344100552684388089096053, 7.65592564221529577871983866374