| L(s) = 1 | + 3-s + 2.82·5-s − 7-s + 9-s − 2·11-s − 2.82·13-s + 2.82·15-s − 19-s − 21-s − 4.82·23-s + 3.00·25-s + 27-s − 3.65·29-s − 1.17·31-s − 2·33-s − 2.82·35-s − 6.48·37-s − 2.82·39-s − 7.65·41-s − 8·43-s + 2.82·45-s − 2.82·47-s + 49-s + 7.65·53-s − 5.65·55-s − 57-s − 1.65·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.26·5-s − 0.377·7-s + 0.333·9-s − 0.603·11-s − 0.784·13-s + 0.730·15-s − 0.229·19-s − 0.218·21-s − 1.00·23-s + 0.600·25-s + 0.192·27-s − 0.679·29-s − 0.210·31-s − 0.348·33-s − 0.478·35-s − 1.06·37-s − 0.452·39-s − 1.19·41-s − 1.21·43-s + 0.421·45-s − 0.412·47-s + 0.142·49-s + 1.05·53-s − 0.762·55-s − 0.132·57-s − 0.215·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 8.82T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−7.66680224764490507555531813850, −6.96713868826347347861929559649, −6.28300535241647708406574508837, −5.49782875112792288550556337822, −4.96820009889317128377389393233, −3.90136767644913125721922483917, −3.05744508154651150166606782529, −2.22001250128405304542165422325, −1.69055406868545000811437621423, 0,
1.69055406868545000811437621423, 2.22001250128405304542165422325, 3.05744508154651150166606782529, 3.90136767644913125721922483917, 4.96820009889317128377389393233, 5.49782875112792288550556337822, 6.28300535241647708406574508837, 6.96713868826347347861929559649, 7.66680224764490507555531813850
