| L(s) = 1 | − 1.41i·3-s − 5-s − 0.732·7-s + 0.999·9-s − 5.27i·11-s − 1.46·13-s + 1.41i·15-s − 6.31i·17-s + 4.24i·19-s + 1.03i·21-s + 8.19·23-s + 25-s − 5.65i·27-s + (−5.19 + 1.41i)29-s + 4.24i·31-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s − 0.447·5-s − 0.276·7-s + 0.333·9-s − 1.59i·11-s − 0.406·13-s + 0.365i·15-s − 1.53i·17-s + 0.973i·19-s + 0.225i·21-s + 1.70·23-s + 0.200·25-s − 1.08i·27-s + (−0.964 + 0.262i)29-s + 0.762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074024577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074024577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + (5.19 - 1.41i)T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 6.31iT - 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 8.76iT - 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 8.38iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 3.10iT - 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 1.13iT - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + 2.07iT - 89T^{2} \) |
| 97 | \( 1 + 7.34iT - 97T^{2} \) |
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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
−8.646326429080559545066155399714, −7.67529897030406957215542048350, −7.20795023212027237653289138414, −6.47326237654290066674501015379, −5.55728646963653702755526927768, −4.72728287401429930062566816235, −3.47735647062810129913086335722, −2.87627109242000962103174978965, −1.44546909418293370403047546921, −0.38112875905549391131519117578,
1.55886991044880893217265483730, 2.79258800791675223531111938517, 3.88093688066149253548001911777, 4.50268838607527986336042497483, 5.10249504617179500395950285419, 6.30836780544580400354980643314, 7.18078314200113158957783414528, 7.64573223523625302878571476942, 8.782473678187901703555732287381, 9.517524711980359789882062516417
