| L(s) = 1 | + (1.5 − 0.866i)5-s + (0.495 − 0.857i)7-s + (−1.81 − 1.05i)11-s + (−5.50 + 3.18i)13-s − 3.81·17-s − 2i·19-s + (−3.55 − 6.15i)23-s + (−1 + 1.73i)25-s + (−7.22 − 4.17i)29-s + (−1.07 − 1.86i)31-s − 1.71i·35-s + 4.62i·37-s + (0.408 + 0.707i)41-s + (1.97 + 1.14i)43-s + (3.39 − 5.87i)47-s + ⋯ |
| L(s) = 1 | + (0.670 − 0.387i)5-s + (0.187 − 0.324i)7-s + (−0.548 − 0.316i)11-s + (−1.52 + 0.882i)13-s − 0.925·17-s − 0.458i·19-s + (−0.740 − 1.28i)23-s + (−0.200 + 0.346i)25-s + (−1.34 − 0.774i)29-s + (−0.193 − 0.335i)31-s − 0.290i·35-s + 0.761i·37-s + (0.0637 + 0.110i)41-s + (0.301 + 0.174i)43-s + (0.494 − 0.857i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4722381320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4722381320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.495 + 0.857i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.81 + 1.05i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.50 - 3.18i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (3.55 + 6.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.22 + 4.17i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.07 + 1.86i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.62iT - 37T^{2} \) |
| 41 | \( 1 + (-0.408 - 0.707i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.97 - 1.14i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.39 + 5.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3.14iT - 53T^{2} \) |
| 59 | \( 1 + (10.3 - 5.95i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.22 + 2.43i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.8 - 6.86i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + (-4.54 + 7.86i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.71 - 2.14i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + (6.22 - 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.194854080235526010717110807250, −8.147006043083919588971681057564, −7.37941170200004477728541736747, −6.56477189236265575790080686791, −5.66777978866978361501593053609, −4.78077283175822699772619069516, −4.14960439243332186983486616230, −2.60032021009577996176707392646, −1.90444108670132011625786661770, −0.15570944660206464435760818811,
1.92514954950813893959373423378, 2.56852580149686902239846943858, 3.75765113987698114986897043562, 5.01686160991511317083550397637, 5.53213526381977100277345502095, 6.42834837635709869707970045108, 7.52538640654492122612491712510, 7.81948287075179251243777426980, 9.159895487727148682708980501959, 9.590234588473442585120411873635
