| L(s) = 1 | − i·3-s + (0.648 − 2.13i)5-s − 2.05i·7-s − 9-s − 1.18·11-s + 2.47i·13-s + (−2.13 − 0.648i)15-s − 3.44i·17-s − 8.19·19-s − 2.05·21-s + 2.07i·23-s + (−4.15 − 2.77i)25-s + i·27-s − 5.48·29-s + 31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.290 − 0.957i)5-s − 0.776i·7-s − 0.333·9-s − 0.356·11-s + 0.687i·13-s + (−0.552 − 0.167i)15-s − 0.835i·17-s − 1.88·19-s − 0.448·21-s + 0.432i·23-s + (−0.831 − 0.555i)25-s + 0.192i·27-s − 1.01·29-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7889712054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7889712054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.648 + 2.13i)T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2.05iT - 7T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 - 2.47iT - 13T^{2} \) |
| 17 | \( 1 + 3.44iT - 17T^{2} \) |
| 19 | \( 1 + 8.19T + 19T^{2} \) |
| 23 | \( 1 - 2.07iT - 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 37 | \( 1 + 6.69iT - 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 - 0.537iT - 47T^{2} \) |
| 53 | \( 1 - 5.53iT - 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4.95iT - 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 - 7.03T + 79T^{2} \) |
| 83 | \( 1 - 8.37iT - 83T^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 - 7.77iT - 97T^{2} \) |
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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
−9.004687553118182718017135030393, −7.87034702641513383457578668444, −7.39111719502771555034712324505, −6.39270354220516517405318849784, −5.67383817353359856205541064475, −4.60565274078883622326966491860, −4.00771383058507950156108252422, −2.49778370922583530753494705061, −1.54039046589140871967752967790, −0.27087748394258484686565800762,
2.05077620380243188921191856076, 2.81442494104815256490430777429, 3.79640599366484689682903842437, 4.76827308897878076977538091049, 5.94585697032650422345572544865, 6.13717472149070831338167779204, 7.32890686112187187174526429322, 8.243807033561890207030817788644, 8.858247873898819841316945269812, 9.774875249484793802525954305747
