| L(s) = 1 | + (−20.3 − 20.3i)3-s + (−76.9 + 76.9i)7-s + 588. i·9-s + 556. i·11-s + (−141. + 141. i)13-s + (477. + 477. i)17-s − 1.60e3·19-s + 3.13e3·21-s + (−346. − 346. i)23-s + (7.03e3 − 7.03e3i)27-s + 7.48e3i·29-s − 7.92e3i·31-s + (1.13e4 − 1.13e4i)33-s + (−3.32e3 − 3.32e3i)37-s + 5.76e3·39-s + ⋯ |
| L(s) = 1 | + (−1.30 − 1.30i)3-s + (−0.593 + 0.593i)7-s + 2.41i·9-s + 1.38i·11-s + (−0.231 + 0.231i)13-s + (0.400 + 0.400i)17-s − 1.02·19-s + 1.55·21-s + (−0.136 − 0.136i)23-s + (1.85 − 1.85i)27-s + 1.65i·29-s − 1.48i·31-s + (1.81 − 1.81i)33-s + (−0.399 − 0.399i)37-s + 0.606·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1342575147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1342575147\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (20.3 + 20.3i)T + 243iT^{2} \) |
| 7 | \( 1 + (76.9 - 76.9i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 556. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (141. - 141. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-477. - 477. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (346. + 346. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 7.48e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.92e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (3.32e3 + 3.32e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.87e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (8.25e3 + 8.25e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (5.09e3 - 5.09e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.94e4 - 1.94e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 108.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (2.89e4 - 2.89e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 982. iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.15e4 + 2.15e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 9.38e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-9.45e3 - 9.45e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 8.48e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.22e5 + 1.22e5i)T + 8.58e9iT^{2} \) |
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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
−10.30530485128193357518904259079, −9.198059215428521849354589956131, −7.86741949094901110055217351339, −7.07316232835476212714052203878, −6.31962791453152714911032360478, −5.52227792975774301385443886646, −4.42190230175318731077389308044, −2.42517383114163919975628156614, −1.52232250054447050691835868233, −0.05924374916914969134820585636,
0.73577286294649735242541365981, 3.17983245505752547553520893053, 4.03908381064120891193404169804, 5.06710183246875013374100473507, 6.00092505394084042293592536157, 6.66809590133797030805912951403, 8.222775841810278123392753302532, 9.386768663983464631734804268179, 10.10616002502234922029678845421, 10.80080936595656899479284085493
