| L(s) = 1 | − i·2-s + 0.618i·3-s − 4-s + 2.23·5-s + 0.618·6-s + 4.85i·7-s + i·8-s + 2.61·9-s − 2.23i·10-s − 3.38·11-s − 0.618i·12-s − 0.381i·13-s + 4.85·14-s + 1.38i·15-s + 16-s − 5.85i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.356i·3-s − 0.5·4-s + 0.999·5-s + 0.252·6-s + 1.83i·7-s + 0.353i·8-s + 0.872·9-s − 0.707i·10-s − 1.01·11-s − 0.178i·12-s − 0.105i·13-s + 1.29·14-s + 0.356i·15-s + 0.250·16-s − 1.41i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 - 2.23T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.618iT - 3T^{2} \) |
| 7 | \( 1 - 4.85iT - 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 + 0.381iT - 13T^{2} \) |
| 17 | \( 1 + 5.85iT - 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 - 3.70iT - 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 + 6.76iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 - 0.763iT - 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 - 7.52iT - 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 - 5.70iT - 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 + 16.0iT - 97T^{2} \) |
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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
−12.20603725897311012268859178484, −11.28836633852854281732621410812, −10.07175209376578307857858067620, −9.506710814602415565983123694025, −8.739100997259001296618798265342, −7.21901164642356176836931676655, −5.42614038066760378104228618650, −5.20972006853873221168916111520, −3.09290535581912478840257014229, −2.03867292215223658109292997604,
1.41485660603361830762438369361, 3.72591792737666733580355794781, 5.02263458193672447892957030321, 6.22090002616129268893278144765, 7.32451563933513431550051456815, 7.76480017470726334006662736190, 9.416488223765838367064657334144, 10.20524524408134718734106209255, 10.89157441437072401433054066802, 12.83657326647230114823033878325
