| L(s) = 1 | − 5.26·3-s − 12.1i·7-s + 18.7·9-s − 4.59·11-s − 19.9·13-s + 4.10i·17-s + (17.9 − 6.30i)19-s + 64.0i·21-s + 35.2i·23-s − 51.1·27-s + 16.4i·29-s − 4.01i·31-s + 24.2·33-s + 10.6·37-s + 104.·39-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 1.73i·7-s + 2.08·9-s − 0.417·11-s − 1.53·13-s + 0.241i·17-s + (0.943 − 0.331i)19-s + 3.05i·21-s + 1.53i·23-s − 1.89·27-s + 0.568i·29-s − 0.129i·31-s + 0.733·33-s + 0.286·37-s + 2.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6015234781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6015234781\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-17.9 + 6.30i)T \) |
| good | 3 | \( 1 + 5.26T + 9T^{2} \) |
| 7 | \( 1 + 12.1iT - 49T^{2} \) |
| 11 | \( 1 + 4.59T + 121T^{2} \) |
| 13 | \( 1 + 19.9T + 169T^{2} \) |
| 17 | \( 1 - 4.10iT - 289T^{2} \) |
| 23 | \( 1 - 35.2iT - 529T^{2} \) |
| 29 | \( 1 - 16.4iT - 841T^{2} \) |
| 31 | \( 1 + 4.01iT - 961T^{2} \) |
| 37 | \( 1 - 10.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 22.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 23.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 97.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.31T + 4.48e3T^{2} \) |
| 71 | \( 1 - 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.02iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 101.T + 9.40e3T^{2} \) |
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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
−9.438266986007141318133382222754, −7.64457544106131704648658450363, −7.41556121787770820531422127021, −6.71450840393385828537589938491, −5.71519708844713860633726924616, −4.95633047242935171628737172299, −4.42255485804386214729902269995, −3.29817409075665195061024240894, −1.51279663532665093474674428328, −0.53832188834347957547397711039,
0.39613572190525213901193141745, 1.97970865856550537422579543959, 2.94207220977769336966411972292, 4.76171312939534321843462107955, 4.95126597748246541025924108188, 5.86992830751628270055495303712, 6.34537616142782283716304474109, 7.33308732433212753886251392074, 8.158430149382888742928491381192, 9.328461413261818611003466618945
