L(s) = 1 | − 2.54·2-s + 4.46·4-s + 0.268·5-s + 2.27·7-s − 6.27·8-s − 0.682·10-s + 11-s + 2.39·13-s − 5.78·14-s + 7.03·16-s + 6.76·17-s + 3.70·19-s + 1.19·20-s − 2.54·22-s + 8.19·23-s − 4.92·25-s − 6.08·26-s + 10.1·28-s − 6.34·29-s + 8.61·31-s − 5.32·32-s − 17.2·34-s + 0.609·35-s + 7.70·37-s − 9.43·38-s − 1.68·40-s + 2.20·41-s + ⋯ |
L(s) = 1 | − 1.79·2-s + 2.23·4-s + 0.119·5-s + 0.859·7-s − 2.21·8-s − 0.215·10-s + 0.301·11-s + 0.663·13-s − 1.54·14-s + 1.75·16-s + 1.64·17-s + 0.851·19-s + 0.268·20-s − 0.542·22-s + 1.70·23-s − 0.985·25-s − 1.19·26-s + 1.92·28-s − 1.17·29-s + 1.54·31-s − 0.941·32-s − 2.95·34-s + 0.103·35-s + 1.26·37-s − 1.53·38-s − 0.266·40-s + 0.345·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.309540076\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.309540076\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 5 | \( 1 - 0.268T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 6.76T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 6.34T + 29T^{2} \) |
| 31 | \( 1 - 8.61T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 - 5.10T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 67 | \( 1 - 0.912T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 7.24T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.110124545850662382544026995211, −7.53120198440436120997517070487, −7.15906734608578008006152238273, −6.01349726732227944409479490760, −5.59586415228133592146475639803, −4.42140661777193592475386120913, −3.29613919052178244532181988757, −2.45848553839438829971140786494, −1.29632229559408550294035536398, −0.967198034930396598429626354649,
0.967198034930396598429626354649, 1.29632229559408550294035536398, 2.45848553839438829971140786494, 3.29613919052178244532181988757, 4.42140661777193592475386120913, 5.59586415228133592146475639803, 6.01349726732227944409479490760, 7.15906734608578008006152238273, 7.53120198440436120997517070487, 8.110124545850662382544026995211