L(s) = 1 | − 2.61·2-s + 3-s + 4.83·4-s − 4.09·5-s − 2.61·6-s − 7-s − 7.42·8-s + 9-s + 10.7·10-s + 5.70·11-s + 4.83·12-s + 2.61·14-s − 4.09·15-s + 9.73·16-s − 4.37·17-s − 2.61·18-s + 3.61·19-s − 19.8·20-s − 21-s − 14.9·22-s + 1.52·23-s − 7.42·24-s + 11.7·25-s + 27-s − 4.83·28-s − 8.65·29-s + 10.7·30-s + ⋯ |
L(s) = 1 | − 1.84·2-s + 0.577·3-s + 2.41·4-s − 1.83·5-s − 1.06·6-s − 0.377·7-s − 2.62·8-s + 0.333·9-s + 3.38·10-s + 1.71·11-s + 1.39·12-s + 0.698·14-s − 1.05·15-s + 2.43·16-s − 1.06·17-s − 0.616·18-s + 0.828·19-s − 4.43·20-s − 0.218·21-s − 3.17·22-s + 0.318·23-s − 1.51·24-s + 2.35·25-s + 0.192·27-s − 0.914·28-s − 1.60·29-s + 1.95·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5659714693\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5659714693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 - 0.399T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 4.72T + 53T^{2} \) |
| 59 | \( 1 + 3.98T + 59T^{2} \) |
| 61 | \( 1 + 1.11T + 61T^{2} \) |
| 67 | \( 1 + 8.84T + 67T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 + 6.08T + 73T^{2} \) |
| 79 | \( 1 + 0.225T + 79T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.698027437225903819331208178620, −7.894538215232472702898098247613, −7.29712653171809954760909768023, −6.97595865073584950857171800003, −6.05059584370492565362935272019, −4.35924911025923201929895354203, −3.68968335000930643753940506876, −2.88265538595874532965123091185, −1.61927155887409746827601164631, −0.58292636246237153998982297317,
0.58292636246237153998982297317, 1.61927155887409746827601164631, 2.88265538595874532965123091185, 3.68968335000930643753940506876, 4.35924911025923201929895354203, 6.05059584370492565362935272019, 6.97595865073584950857171800003, 7.29712653171809954760909768023, 7.894538215232472702898098247613, 8.698027437225903819331208178620