| L(s) = 1 | + i·3-s − i·5-s − 1.44·7-s − 9-s + 0.540·11-s + 4.25·13-s + 15-s + 7.14i·17-s + 0.941·19-s − 1.44i·21-s + (−2.91 + 3.80i)23-s − 25-s − i·27-s + 1.26·29-s − 3.15i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.547·7-s − 0.333·9-s + 0.162·11-s + 1.18·13-s + 0.258·15-s + 1.73i·17-s + 0.215·19-s − 0.315i·21-s + (−0.607 + 0.794i)23-s − 0.200·25-s − 0.192i·27-s + 0.234·29-s − 0.566i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.373535851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373535851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.91 - 3.80i)T \) |
| good | 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 0.540T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 - 7.14iT - 17T^{2} \) |
| 19 | \( 1 - 0.941T + 19T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 + 3.15iT - 31T^{2} \) |
| 37 | \( 1 + 1.30iT - 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 + 6.61iT - 47T^{2} \) |
| 53 | \( 1 - 7.54iT - 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.00iT - 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 - 8.07iT - 71T^{2} \) |
| 73 | \( 1 - 1.91T + 73T^{2} \) |
| 79 | \( 1 - 3.99T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−8.292808135972007856432840949838, −7.987122818139542790713725593072, −6.80562786466734270041208032397, −6.02826465345929568627288493266, −5.69125983466893801766083788574, −4.61745186185509812851573376796, −3.78819845745507935791378255260, −3.47776065365911687549394976365, −2.13136100423209263256536597533, −1.11719394784804196515584309658,
0.39409955507853712654031843210, 1.49643772340186863190440857141, 2.68894781932468848538548288678, 3.18804095782305413237674117729, 4.18284472372387886632543716318, 5.09340366879950250866697621853, 6.07420566739895504516090318099, 6.44625006041617608275587444685, 7.20319374635661470751232700802, 7.78720312344131696283374767007
