| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s − 10-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 20-s + 21-s − 24-s + 26-s + 27-s − 28-s + 30-s + 2·31-s + 32-s − 34-s + 35-s − 37-s − 39-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s − 10-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 20-s + 21-s − 24-s + 26-s + 27-s − 28-s + 30-s + 2·31-s + 32-s − 34-s + 35-s − 37-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6434240793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6434240793\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−13.81741639926511180893815804249, −12.91148629931192100169279338687, −11.87970019424135056761325974975, −11.33427763736527423692501678920, −10.28455626039788331453937391311, −8.374845901588246109940636090068, −6.80322656561684872784900392938, −6.08438169252065874846073663176, −4.62506222520716784109149308317, −3.31991104814500528755495843356,
3.31991104814500528755495843356, 4.62506222520716784109149308317, 6.08438169252065874846073663176, 6.80322656561684872784900392938, 8.374845901588246109940636090068, 10.28455626039788331453937391311, 11.33427763736527423692501678920, 11.87970019424135056761325974975, 12.91148629931192100169279338687, 13.81741639926511180893815804249
