L(s) = 1 | + 2.50·2-s − 16.8·3-s − 25.7·4-s − 42.2·6-s − 67.4·7-s − 144.·8-s + 42.2·9-s + 81.3·11-s + 434.·12-s − 1.05e3·13-s − 168.·14-s + 462.·16-s − 251.·17-s + 105.·18-s + 1.61e3·19-s + 1.13e3·21-s + 203.·22-s + 32.7·23-s + 2.43e3·24-s − 2.64e3·26-s + 3.39e3·27-s + 1.73e3·28-s − 2.58e3·29-s − 7.20e3·31-s + 5.77e3·32-s − 1.37e3·33-s − 628.·34-s + ⋯ |
L(s) = 1 | + 0.441·2-s − 1.08·3-s − 0.804·4-s − 0.478·6-s − 0.520·7-s − 0.797·8-s + 0.173·9-s + 0.202·11-s + 0.871·12-s − 1.73·13-s − 0.229·14-s + 0.452·16-s − 0.210·17-s + 0.0768·18-s + 1.02·19-s + 0.563·21-s + 0.0896·22-s + 0.0129·23-s + 0.864·24-s − 0.767·26-s + 0.895·27-s + 0.418·28-s − 0.570·29-s − 1.34·31-s + 0.997·32-s − 0.219·33-s − 0.0932·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
good | 2 | \( 1 - 2.50T + 32T^{2} \) |
| 3 | \( 1 + 16.8T + 243T^{2} \) |
| 7 | \( 1 + 67.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 81.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 251.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 32.7T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.17e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.55e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.69e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.20e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5T + 8.58e9T^{2} \) |
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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
−9.015132023127760449943464090876, −7.73215010365809294394074612582, −6.92541509657053697024376615111, −5.88409857686895811652495111569, −5.32300049184609891623809906497, −4.61639702607657774982812692328, −3.58826988474829778547341695148, −2.49293739029635753778106291564, −0.77184713408178335094970326694, 0,
0.77184713408178335094970326694, 2.49293739029635753778106291564, 3.58826988474829778547341695148, 4.61639702607657774982812692328, 5.32300049184609891623809906497, 5.88409857686895811652495111569, 6.92541509657053697024376615111, 7.73215010365809294394074612582, 9.015132023127760449943464090876