| L(s) = 1 | + 5i·5-s + (−2 − 2i)7-s − 8·11-s + (3 − 3i)13-s + (−7 − 7i)17-s + 20i·19-s + (−2 + 2i)23-s − 25·25-s − 40i·29-s − 52·31-s + (10 − 10i)35-s + (−3 − 3i)37-s + 8·41-s + (42 − 42i)43-s + (−18 − 18i)47-s + ⋯ |
| L(s) = 1 | + i·5-s + (−0.285 − 0.285i)7-s − 0.727·11-s + (0.230 − 0.230i)13-s + (−0.411 − 0.411i)17-s + 1.05i·19-s + (−0.0869 + 0.0869i)23-s − 25-s − 1.37i·29-s − 1.67·31-s + (0.285 − 0.285i)35-s + (−0.0810 − 0.0810i)37-s + 0.195·41-s + (0.976 − 0.976i)43-s + (−0.382 − 0.382i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1139154368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1139154368\) |
| \(L(2)\) |
|
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 | \( 1 - 5iT \) |
| good | 7 | \( 1 + (2 + 2i)T + 49iT^{2} \) |
| 11 | \( 1 + 8T + 121T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 169iT^{2} \) |
| 17 | \( 1 + (7 + 7i)T + 289iT^{2} \) |
| 19 | \( 1 - 20iT - 361T^{2} \) |
| 23 | \( 1 + (2 - 2i)T - 529iT^{2} \) |
| 29 | \( 1 + 40iT - 841T^{2} \) |
| 31 | \( 1 + 52T + 961T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-42 + 42i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (18 + 18i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (53 - 53i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 20iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48T + 3.72e3T^{2} \) |
| 67 | \( 1 + (62 + 62i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 28T + 5.04e3T^{2} \) |
| 73 | \( 1 + (47 - 47i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + (-18 + 18i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 80iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (63 + 63i)T + 9.40e3iT^{2} \) |
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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
−10.04587011064118405111832061442, −9.104992412260222925052665047700, −7.87440398540535583352577318744, −7.34518338515597913000953135167, −6.28762860431732638331318711526, −5.52568223130309112187035362729, −4.10963782785023855474001651569, −3.18405395875085063176751751232, −2.05467038507331324426782517180, −0.03800039163419891269533218896,
1.56505578793726755450831767741, 2.93270162111964406082395065266, 4.26501136940070245675014862329, 5.13885129083547529186100376375, 5.99794477239743205867499296868, 7.12576294826416921730352317234, 8.087383178800547198049523989369, 8.986122776586366939708598110459, 9.419272979089995967602279730140, 10.66536886304402337565507196080
