L(s) = 1 | − 3.42i·3-s − 9.18·7-s − 2.73·9-s + 11.4·11-s + 10.1i·13-s + 4.19·17-s + (10.8 − 15.6i)19-s + 31.4i·21-s + 0.952·23-s − 21.4i·27-s − 9.43i·29-s + 8.52i·31-s − 39.1i·33-s − 20.0i·37-s + 34.6·39-s + ⋯ |
L(s) = 1 | − 1.14i·3-s − 1.31·7-s − 0.303·9-s + 1.03·11-s + 0.778i·13-s + 0.246·17-s + (0.568 − 0.822i)19-s + 1.49i·21-s + 0.0414·23-s − 0.795i·27-s − 0.325i·29-s + 0.275i·31-s − 1.18i·33-s − 0.540i·37-s + 0.889·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.595445573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.595445573\) |
\(L(2)\) |
|
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 | \( 1 + (-10.8 + 15.6i)T \) |
good | 3 | \( 1 + 3.42iT - 9T^{2} \) |
| 7 | \( 1 + 9.18T + 49T^{2} \) |
| 11 | \( 1 - 11.4T + 121T^{2} \) |
| 13 | \( 1 - 10.1iT - 169T^{2} \) |
| 17 | \( 1 - 4.19T + 289T^{2} \) |
| 23 | \( 1 - 0.952T + 529T^{2} \) |
| 29 | \( 1 + 9.43iT - 841T^{2} \) |
| 31 | \( 1 - 8.52iT - 961T^{2} \) |
| 37 | \( 1 + 20.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 61.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 57.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 57.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 32.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.07iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 27.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 60.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 97.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 92.0iT - 9.40e3T^{2} \) |
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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
−8.900385575469164069204365865280, −7.73344419362170581960471655799, −7.06518646021704990991614682461, −6.49227796819244999925486508501, −5.96763022996207258976304472286, −4.59395154753339811820481003823, −3.61682645692307902391843578339, −2.64299050239022561270567925858, −1.52311077650644271270507602034, −0.48864343439614198060089530918,
1.06001296138995302287904806363, 2.77073338386557353831792377900, 3.66338403221633536945825237986, 4.08763732606172822171135323475, 5.32953477259042143547049723639, 5.97394830189443966651556211285, 6.88513806567456164047337131324, 7.71451331038889224812896909175, 8.928001048588117598795416473624, 9.324618691803068771495990275216