| L(s) = 1 | − i·2-s + i·3-s + 4-s + (1 + 2i)5-s + 6-s + 2i·7-s − 3i·8-s − 9-s + (2 − i)10-s + i·12-s + 4i·13-s + 2·14-s + (−2 + i)15-s − 16-s − 2i·17-s + i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + (0.447 + 0.894i)5-s + 0.408·6-s + 0.755i·7-s − 1.06i·8-s − 0.333·9-s + (0.632 − 0.316i)10-s + 0.288i·12-s + 1.10i·13-s + 0.534·14-s + (−0.516 + 0.258i)15-s − 0.250·16-s − 0.485i·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66434 + 0.392898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66434 + 0.392898i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−11.46703146214489623876729206357, −10.30850117503531401580472386251, −9.699035994704016622521748578347, −8.858554209804972962442823422743, −7.34519022737667832592625913677, −6.49736812590098311652648057645, −5.54235187190553773505443772454, −4.03626721994331605120916190742, −2.90221470476548853027593523716, −2.01746434099217470805047588555,
1.18931696640093760457797633098, 2.70725007613195789371249212216, 4.48179666436169288321200462777, 5.68843211675051030193084740788, 6.29898920478888041882346927315, 7.50945056583642494634706499276, 8.015895656487389401265565457087, 9.002089539462079661753590258004, 10.26811213835576722195467255803, 11.00608691485231665295111499680
