L(s) = 1 | + 2-s − 3.01·3-s + 4-s + 5-s − 3.01·6-s + 2.60·7-s + 8-s + 6.08·9-s + 10-s + 2.96·11-s − 3.01·12-s − 13-s + 2.60·14-s − 3.01·15-s + 16-s + 1.73·17-s + 6.08·18-s + 0.404·19-s + 20-s − 7.86·21-s + 2.96·22-s + 8.70·23-s − 3.01·24-s + 25-s − 26-s − 9.31·27-s + 2.60·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.74·3-s + 0.5·4-s + 0.447·5-s − 1.23·6-s + 0.985·7-s + 0.353·8-s + 2.02·9-s + 0.316·10-s + 0.894·11-s − 0.870·12-s − 0.277·13-s + 0.697·14-s − 0.778·15-s + 0.250·16-s + 0.421·17-s + 1.43·18-s + 0.0926·19-s + 0.223·20-s − 1.71·21-s + 0.632·22-s + 1.81·23-s − 0.615·24-s + 0.200·25-s − 0.196·26-s − 1.79·27-s + 0.492·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.434161819\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.434161819\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 3.01T + 3T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 0.404T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 + 8.52T + 41T^{2} \) |
| 43 | \( 1 + 0.800T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 59 | \( 1 + 2.36T + 59T^{2} \) |
| 61 | \( 1 + 8.34T + 61T^{2} \) |
| 67 | \( 1 + 0.173T + 67T^{2} \) |
| 71 | \( 1 + 0.460T + 71T^{2} \) |
| 73 | \( 1 - 1.98T + 73T^{2} \) |
| 79 | \( 1 - 0.457T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 3.31T + 97T^{2} \) |
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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
−8.299175971827076988145442867529, −7.34074487584682874876307928513, −6.65914184898620827951569806798, −6.19570593911903494593876530090, −5.26512207049678023682252862074, −4.93408686257952917120959701689, −4.28630338622001720813034374216, −3.07328232453547136503686914844, −1.67067941269524647013771938929, −0.975708623473008345219339238701,
0.975708623473008345219339238701, 1.67067941269524647013771938929, 3.07328232453547136503686914844, 4.28630338622001720813034374216, 4.93408686257952917120959701689, 5.26512207049678023682252862074, 6.19570593911903494593876530090, 6.65914184898620827951569806798, 7.34074487584682874876307928513, 8.299175971827076988145442867529