| L(s) = 1 | − i·3-s − i·5-s − 0.482·7-s − 9-s − 1.10·11-s + 2.25·13-s − 15-s − 6.03i·17-s + 1.48·19-s + 0.482i·21-s + (4.55 − 1.50i)23-s − 25-s + i·27-s − 7.68·29-s + 5.12i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.182·7-s − 0.333·9-s − 0.332·11-s + 0.625·13-s − 0.258·15-s − 1.46i·17-s + 0.340·19-s + 0.105i·21-s + (0.949 − 0.314i)23-s − 0.200·25-s + 0.192i·27-s − 1.42·29-s + 0.920i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.204193085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204193085\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 23 | \( 1 + (-4.55 + 1.50i)T \) |
| good | 7 | \( 1 + 0.482T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 + 6.03iT - 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 - 5.12iT - 31T^{2} \) |
| 37 | \( 1 + 5.83iT - 37T^{2} \) |
| 41 | \( 1 - 1.27T + 41T^{2} \) |
| 43 | \( 1 - 8.46T + 43T^{2} \) |
| 47 | \( 1 + 11.2iT - 47T^{2} \) |
| 53 | \( 1 - 8.81iT - 53T^{2} \) |
| 59 | \( 1 + 8.08iT - 59T^{2} \) |
| 61 | \( 1 - 0.0392iT - 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 - 1.25iT - 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 6.85iT - 89T^{2} \) |
| 97 | \( 1 - 0.330iT - 97T^{2} \) |
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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
−7.72176555234247378839431194623, −7.17413715094601519925375265835, −6.53857686975647509519586094621, −5.50217175417383166772544756946, −5.19004186333119214138433743852, −4.10893437493273155662325172291, −3.20190101526348462244816435583, −2.39550749308442522568449245317, −1.30447436318360766801696418597, −0.33103828136453240524683841560,
1.30346048427862691993128791674, 2.44592612954395511073839006339, 3.36820908662381538237592777439, 3.91691114296765137363249907189, 4.78865541003802237759281076723, 5.76813961667908951787597875886, 6.11794399938805080794164923522, 7.07766627562421600282893265681, 7.82882003720763578040832251534, 8.440914321235256399779489952641
