| L(s) = 1 | + 3.95·2-s − 3.98·3-s − 16.3·4-s − 15.7·6-s + 105.·7-s − 191.·8-s − 227.·9-s − 502.·11-s + 65.2·12-s − 941.·13-s + 415.·14-s − 232.·16-s − 1.44e3·17-s − 897.·18-s − 1.55e3·19-s − 418.·21-s − 1.98e3·22-s − 1.62e3·23-s + 762.·24-s − 3.72e3·26-s + 1.87e3·27-s − 1.71e3·28-s + 4.58e3·29-s − 7.28e3·31-s + 5.20e3·32-s + 2.00e3·33-s − 5.70e3·34-s + ⋯ |
| L(s) = 1 | + 0.698·2-s − 0.255·3-s − 0.511·4-s − 0.178·6-s + 0.810·7-s − 1.05·8-s − 0.934·9-s − 1.25·11-s + 0.130·12-s − 1.54·13-s + 0.566·14-s − 0.226·16-s − 1.21·17-s − 0.653·18-s − 0.989·19-s − 0.207·21-s − 0.875·22-s − 0.638·23-s + 0.270·24-s − 1.08·26-s + 0.494·27-s − 0.414·28-s + 1.01·29-s − 1.36·31-s + 0.897·32-s + 0.320·33-s − 0.847·34-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
| \(L(3)\) |
\(\approx\) |
\(0.09123085170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09123085170\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 3.95T + 32T^{2} \) |
| 3 | \( 1 + 3.98T + 243T^{2} \) |
| 7 | \( 1 - 105.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 502.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 941.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.44e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.72e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.37e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.20e4T + 8.58e9T^{2} \) |
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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
−8.944182110230403015312844511915, −8.433433428941583008973881791370, −7.52669959210641846011643767846, −6.41534823214478288509668777326, −5.37520767609945027941236811238, −4.97814409406944372814966096337, −4.18014419291204467731533986052, −2.83799681755773350892362999556, −2.12215450342357955637256510764, −0.10905829064094743544750891019,
0.10905829064094743544750891019, 2.12215450342357955637256510764, 2.83799681755773350892362999556, 4.18014419291204467731533986052, 4.97814409406944372814966096337, 5.37520767609945027941236811238, 6.41534823214478288509668777326, 7.52669959210641846011643767846, 8.433433428941583008973881791370, 8.944182110230403015312844511915
