| L(s) = 1 | + (−0.0928 − 0.0649i)2-s + (−0.679 − 1.86i)4-s + (−0.584 − 2.15i)5-s + (0.344 + 0.739i)7-s + (−0.116 + 0.436i)8-s + (−0.0860 + 0.238i)10-s + (0.792 + 0.944i)11-s + (−2.08 − 2.97i)13-s + (0.0160 − 0.0910i)14-s + (−3.00 + 2.52i)16-s + (−1.65 − 6.18i)17-s + (−2.78 + 1.60i)19-s + (−3.63 + 2.55i)20-s + (−0.0121 − 0.139i)22-s + (−0.471 − 0.219i)23-s + ⋯ |
| L(s) = 1 | + (−0.0656 − 0.0459i)2-s + (−0.339 − 0.933i)4-s + (−0.261 − 0.965i)5-s + (0.130 + 0.279i)7-s + (−0.0413 + 0.154i)8-s + (−0.0272 + 0.0753i)10-s + (0.238 + 0.284i)11-s + (−0.577 − 0.825i)13-s + (0.00429 − 0.0243i)14-s + (−0.751 + 0.630i)16-s + (−0.402 − 1.50i)17-s + (−0.638 + 0.368i)19-s + (−0.812 + 0.571i)20-s + (−0.00259 − 0.0296i)22-s + (−0.0982 − 0.0458i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.293245 - 0.806813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.293245 - 0.806813i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.584 + 2.15i)T \) |
| good | 2 | \( 1 + (0.0928 + 0.0649i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.344 - 0.739i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.792 - 0.944i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.08 + 2.97i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.65 + 6.18i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.78 - 1.60i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.471 + 0.219i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.754 + 4.28i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.93 - 2.52i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-8.35 + 2.23i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-8.94 - 1.57i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.490 + 5.60i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (2.14 - 1.00i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (4.55 + 4.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (-10.5 - 8.86i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.01 - 1.09i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.77 + 6.84i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (4.14 + 2.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.46 + 1.46i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.54 + 1.68i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (4.76 - 6.80i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-0.689 - 1.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.18 + 0.540i)T + (95.5 + 16.8i)T^{2} \) |
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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
−10.90505602959910623725755170731, −9.759680086060060061441892524006, −9.254345372718449755641583044845, −8.298860234271794642725789025253, −7.22206831714622144480536931298, −5.82409727552272898389387002746, −5.08955746824318917085138769171, −4.18244648983118945479674650694, −2.22441426728625587152292429819, −0.56502198693212961829104857090,
2.35930057351484156081364663766, 3.70160913299168623873546761434, 4.40050586573891627900839110940, 6.15797873145529914520846011527, 7.05868832150700938396815127504, 7.84588080385667320858040264394, 8.782873425912953358421763156956, 9.731205029843848019254323649364, 10.95456850706901811612225100100, 11.39868514943392146592381915627
