L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.227 + 0.151i)3-s + (−0.707 + 0.707i)4-s + (−2.10 − 0.418i)5-s + (0.227 + 0.151i)6-s + (−1.07 + 2.41i)7-s + (0.923 + 0.382i)8-s + (−1.11 + 2.70i)9-s + (0.418 + 2.10i)10-s + (0.290 + 0.194i)11-s + (0.0532 − 0.267i)12-s + (−1.02 + 1.02i)13-s + (2.64 + 0.0704i)14-s + (0.541 − 0.224i)15-s − i·16-s + (3.43 + 2.28i)17-s + ⋯ |
L(s) = 1 | + (−0.270 − 0.653i)2-s + (−0.131 + 0.0875i)3-s + (−0.353 + 0.353i)4-s + (−0.941 − 0.187i)5-s + (0.0926 + 0.0619i)6-s + (−0.407 + 0.913i)7-s + (0.326 + 0.135i)8-s + (−0.373 + 0.900i)9-s + (0.132 + 0.666i)10-s + (0.0877 + 0.0586i)11-s + (0.0153 − 0.0773i)12-s + (−0.285 + 0.285i)13-s + (0.706 + 0.0188i)14-s + (0.139 − 0.0579i)15-s − 0.250i·16-s + (0.832 + 0.554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00699 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00699 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332537 + 0.334871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332537 + 0.334871i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 + (1.07 - 2.41i)T \) |
| 17 | \( 1 + (-3.43 - 2.28i)T \) |
good | 3 | \( 1 + (0.227 - 0.151i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (2.10 + 0.418i)T + (4.61 + 1.91i)T^{2} \) |
| 11 | \( 1 + (-0.290 - 0.194i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (1.02 - 1.02i)T - 13iT^{2} \) |
| 19 | \( 1 + (6.61 - 2.74i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.525 + 0.785i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.856 + 0.170i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-1.57 - 2.35i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (1.73 + 2.59i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-8.32 + 1.65i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.47 - 5.98i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (4.25 - 4.25i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.77 + 6.70i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.263 + 0.635i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (1.93 + 9.74i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 7.81iT - 67T^{2} \) |
| 71 | \( 1 + (5.40 + 8.09i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-8.54 - 1.69i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-11.1 - 7.46i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-6.09 - 14.7i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (0.0673 + 0.0673i)T + 89iT^{2} \) |
| 97 | \( 1 + (3.00 - 15.1i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−12.34915891205960553597817490095, −11.43337990104711478960132302949, −10.61874812525001935162573826213, −9.552688078226156468822281951812, −8.419162470658074770215845780615, −7.87255668277036137456971638666, −6.23421304725162534116487182657, −4.90844263743359965086438785029, −3.68835176918771220013685340086, −2.22860121427435547112852174014,
0.41529270521331604594225291356, 3.36014210677795055207901443749, 4.46557131535159291825002716212, 6.02691122670276895359752744687, 7.00245504874707871446860002498, 7.74587397114767395732411581118, 8.859473817188223827833552624698, 9.900622820245212954922706504573, 10.91008157552272533401901044581, 11.87648457431735856580334394538