| L(s) = 1 | + 1.88·2-s − 3-s + 1.55·4-s − 5-s − 1.88·6-s + 2.76·7-s − 0.841·8-s + 9-s − 1.88·10-s + 0.966·11-s − 1.55·12-s − 2.43·13-s + 5.21·14-s + 15-s − 4.69·16-s + 7.28·17-s + 1.88·18-s − 7.75·19-s − 1.55·20-s − 2.76·21-s + 1.82·22-s − 1.67·23-s + 0.841·24-s + 25-s − 4.59·26-s − 27-s + 4.29·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.577·3-s + 0.776·4-s − 0.447·5-s − 0.769·6-s + 1.04·7-s − 0.297·8-s + 0.333·9-s − 0.596·10-s + 0.291·11-s − 0.448·12-s − 0.675·13-s + 1.39·14-s + 0.258·15-s − 1.17·16-s + 1.76·17-s + 0.444·18-s − 1.77·19-s − 0.347·20-s − 0.603·21-s + 0.388·22-s − 0.349·23-s + 0.171·24-s + 0.200·25-s − 0.900·26-s − 0.192·27-s + 0.811·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 - 0.966T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 7.28T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 + 0.559T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 8.45T + 47T^{2} \) |
| 53 | \( 1 - 3.42T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.47025495595102788245652223547, −6.93830588575989955836649804572, −5.93403547714207916822186326988, −5.49073821543481195816361620024, −4.82043580611245479339300488714, −4.15106094249606369662785140191, −3.61720828568574061548050728087, −2.49083620585299639278039338274, −1.52012413147596936775829971541, 0,
1.52012413147596936775829971541, 2.49083620585299639278039338274, 3.61720828568574061548050728087, 4.15106094249606369662785140191, 4.82043580611245479339300488714, 5.49073821543481195816361620024, 5.93403547714207916822186326988, 6.93830588575989955836649804572, 7.47025495595102788245652223547
