| L(s) = 1 | − 4.47i·7-s − 2.23·11-s + 4i·13-s − 7i·17-s + 6.70·19-s + 4.47i·23-s + 4.47·31-s − 2i·37-s − 5·41-s − 8.94i·47-s − 13.0·49-s − 6i·53-s + 8.94·59-s + 10·61-s + 2.23i·67-s + ⋯ |
| L(s) = 1 | − 1.69i·7-s − 0.674·11-s + 1.10i·13-s − 1.69i·17-s + 1.53·19-s + 0.932i·23-s + 0.803·31-s − 0.328i·37-s − 0.780·41-s − 1.30i·47-s − 1.85·49-s − 0.824i·53-s + 1.16·59-s + 1.28·61-s + 0.273i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584453251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584453251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.51632126960610043415229032097, −7.07285863866384932498644139999, −6.62989510940219012964138335644, −5.26463914749291873568210757351, −5.01829961303281057425261312630, −3.99259519659740165898230981111, −3.44713520898222059297840299683, −2.46638329762792549391810432918, −1.29348053404205404734656024124, −0.43061105971833427041287858923,
1.12233395391731144027394747420, 2.30861678740120204224271111606, 2.85720539381368682477026792647, 3.65561792616361294338861589601, 4.83719862463972772640668932897, 5.42203298465228833083737670731, 5.92542995203487407829889541354, 6.58268991199515317411171588345, 7.68329358886812684893898113016, 8.253529210944521344842222668456
