| L(s) = 1 | − 1.53·5-s − 0.585i·7-s + 5.22i·11-s − 5.22i·13-s + 1.41i·17-s + 3.06·19-s + 0.828·23-s − 2.65·25-s − 5.86·29-s − 6.24i·31-s + 0.896i·35-s + 3.06i·37-s − 3.75i·41-s − 4.32·43-s + 8.82·47-s + ⋯ |
| L(s) = 1 | − 0.684·5-s − 0.221i·7-s + 1.57i·11-s − 1.44i·13-s + 0.342i·17-s + 0.702·19-s + 0.172·23-s − 0.531·25-s − 1.08·29-s − 1.12i·31-s + 0.151i·35-s + 0.503i·37-s − 0.586i·41-s − 0.660·43-s + 1.28·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282069460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282069460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 0.585iT - 7T^{2} \) |
| 11 | \( 1 - 5.22iT - 11T^{2} \) |
| 13 | \( 1 + 5.22iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 + 6.24iT - 31T^{2} \) |
| 37 | \( 1 - 3.06iT - 37T^{2} \) |
| 41 | \( 1 + 3.75iT - 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 5.86T + 53T^{2} \) |
| 59 | \( 1 + 4.32iT - 59T^{2} \) |
| 61 | \( 1 - 7.39iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 2.92iT - 79T^{2} \) |
| 83 | \( 1 - 9.55iT - 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.958190226988600618152003457230, −7.51778553837004507722433427048, −7.06664282073398688094724210960, −5.88379393484163157257552266165, −5.30185635535806941669293672684, −4.33696209351185927915768754311, −3.77546381797512104403696223412, −2.79313889536515809567635511860, −1.77462599007617442061477646059, −0.45873411784773335247424215848,
0.867626002596034630020483506329, 2.09068133914807457610018119613, 3.22777451455141064225712983876, 3.77992640578683780167755869383, 4.66455059586218993269482782183, 5.57543147085921164249328996305, 6.18709471017965459220963391732, 7.13234646725599916295012077346, 7.60366843165938265570277207391, 8.611566555929387392647055470373
