| L(s) = 1 | + 3.00·3-s + (−2.14 + 0.621i)5-s + 7-s + 6.03·9-s − 5.31i·11-s − 4.82i·13-s + (−6.45 + 1.86i)15-s + (0.461 − 4.09i)17-s − 5.70·19-s + 3.00·21-s − 7.79·23-s + (4.22 − 2.66i)25-s + 9.12·27-s − 4.41i·29-s + 6.68i·31-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + (−0.960 + 0.277i)5-s + 0.377·7-s + 2.01·9-s − 1.60i·11-s − 1.33i·13-s + (−1.66 + 0.482i)15-s + (0.111 − 0.993i)17-s − 1.30·19-s + 0.656·21-s − 1.62·23-s + (0.845 − 0.533i)25-s + 1.75·27-s − 0.820i·29-s + 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.518866632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.518866632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.14 - 0.621i)T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + (-0.461 + 4.09i)T \) |
| good | 3 | \( 1 - 3.00T + 3T^{2} \) |
| 11 | \( 1 + 5.31iT - 11T^{2} \) |
| 13 | \( 1 + 4.82iT - 13T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 + 4.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.68iT - 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 + 8.61iT - 41T^{2} \) |
| 43 | \( 1 - 6.39iT - 43T^{2} \) |
| 47 | \( 1 + 1.93iT - 47T^{2} \) |
| 53 | \( 1 - 2.78iT - 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 6.19iT - 61T^{2} \) |
| 67 | \( 1 + 12.8iT - 67T^{2} \) |
| 71 | \( 1 - 0.219iT - 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 8.79T + 89T^{2} \) |
| 97 | \( 1 + 3.66T + 97T^{2} \) |
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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
−8.496184258965698844028346514100, −8.108379605481143717424678380430, −7.73747859077352580902080552307, −6.71814480723064156846972561562, −5.67514532120083141252537898042, −4.44241331942507516749102808326, −3.67039371132709436002241518334, −3.06080514786037330279133536880, −2.28657565144540618521925187630, −0.64987390155147227801225743131,
1.80658136276440364604852853657, 2.16111893430223937714599439493, 3.58952331299983450163854996002, 4.30627269682601091692440370181, 4.52854719211066750927915925520, 6.30806596632088235634244356396, 7.19773475219068652302961545380, 7.77211174192505764317515644843, 8.368495685061802630460366924802, 8.937090048450194736330822665357
