| L(s) = 1 | + (−1.39 − 0.201i)2-s + (1.46 + 2.68i)3-s + (1.91 + 0.564i)4-s + (1.95 + 1.08i)5-s + (−1.51 − 4.05i)6-s + (−0.897 − 1.19i)7-s + (−2.57 − 1.17i)8-s + (−3.44 + 5.36i)9-s + (−2.51 − 1.91i)10-s + (3.30 − 1.50i)11-s + (1.29 + 5.98i)12-s + (3.96 + 2.96i)13-s + (1.01 + 1.85i)14-s + (−0.0620 + 6.84i)15-s + (3.36 + 2.16i)16-s + (1.77 − 0.127i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.847 + 1.55i)3-s + (0.959 + 0.282i)4-s + (0.873 + 0.487i)5-s + (−0.617 − 1.65i)6-s + (−0.339 − 0.453i)7-s + (−0.909 − 0.415i)8-s + (−1.14 + 1.78i)9-s + (−0.794 − 0.606i)10-s + (0.995 − 0.454i)11-s + (0.375 + 1.72i)12-s + (1.09 + 0.823i)13-s + (0.271 + 0.496i)14-s + (−0.0160 + 1.76i)15-s + (0.840 + 0.541i)16-s + (0.431 − 0.0308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0634 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0634 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.959724 + 1.02272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.959724 + 1.02272i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.201i)T \) |
| 5 | \( 1 + (-1.95 - 1.08i)T \) |
| 23 | \( 1 + (4.60 - 1.32i)T \) |
| good | 3 | \( 1 + (-1.46 - 2.68i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (0.897 + 1.19i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-3.30 + 1.50i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-3.96 - 2.96i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-1.77 + 0.127i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (1.64 + 1.89i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (6.19 + 5.37i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.32 + 4.49i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.54 - 7.08i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (-3.94 + 2.53i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (2.89 + 5.30i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (6.17 - 6.17i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.15 + 5.54i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (1.60 + 11.1i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.03 + 0.892i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.64 - 9.77i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (2.37 + 1.08i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (2.60 + 0.186i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.671 - 4.67i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.218 - 1.00i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (2.80 - 9.56i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-18.0 - 3.92i)T + (88.2 + 40.2i)T^{2} \) |
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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
−11.05721391798877505556622028911, −9.973222494648969303630638195459, −9.643365815907799082619332598160, −8.933611061457386719108743112536, −8.088107585368356655368372838540, −6.69524906856439046283083842132, −5.83576024157823010856588485693, −4.01720516168367508861519856562, −3.35093056293644205249956044600, −1.97641729043474308041837598685,
1.23194158176423616554340041805, 2.00832315110998711968650872203, 3.29650477086793954223013454747, 5.85788753462026423866825797550, 6.27689907436080352683003497133, 7.31693534067020249882982818393, 8.228852776357050612210813199395, 8.903002382988308361488578644088, 9.489230984840068894341385487047, 10.64176468573100519965346910129
