| L(s) = 1 | + i·3-s − i·5-s + 5.01·7-s − 9-s + 2.36·11-s + 5.01·13-s + 15-s − 1.91i·17-s − 4.78·19-s + 5.01i·21-s + (−3.28 − 3.49i)23-s − 25-s − i·27-s + 5.75·29-s + 3.94i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 1.89·7-s − 0.333·9-s + 0.713·11-s + 1.39·13-s + 0.258·15-s − 0.464i·17-s − 1.09·19-s + 1.09i·21-s + (−0.684 − 0.728i)23-s − 0.200·25-s − 0.192i·27-s + 1.06·29-s + 0.708i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.918846620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.918846620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (3.28 + 3.49i)T \) |
| good | 7 | \( 1 - 5.01T + 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 + 1.91iT - 17T^{2} \) |
| 19 | \( 1 + 4.78T + 19T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 - 3.94iT - 31T^{2} \) |
| 37 | \( 1 - 8.21iT - 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 - 8.32iT - 47T^{2} \) |
| 53 | \( 1 - 1.72iT - 53T^{2} \) |
| 59 | \( 1 + 4.37iT - 59T^{2} \) |
| 61 | \( 1 + 2.56iT - 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 + 9.36iT - 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + 0.143T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−8.264054275770773781396782623583, −7.80384165425961148788299100201, −6.58959238168595208129879249919, −5.99841337177867699376302058712, −5.10810050898193468357036626516, −4.34376768488842668174905953180, −4.17931271113808748413604171780, −2.84890476137066111940069189181, −1.73377520393335597077342839592, −1.00077625817865363039400886350,
1.01900325138453522487664203437, 1.76697190455577196708410049382, 2.49582341283368432384591883472, 4.04940820083274680494414967574, 4.09415510719226481678962757656, 5.45137743731555079240656033375, 5.97585259966118616649431785488, 6.70110918963404933650964673962, 7.55656851849682311252891256173, 8.093720368379949608785963070634
