| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s + 11-s + 2·13-s − 2·15-s − 2·17-s + 4·21-s − 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s − 33-s − 8·35-s − 6·37-s − 2·39-s − 2·41-s + 2·45-s − 8·47-s + 9·49-s + 2·51-s − 6·53-s + 2·55-s − 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.174·33-s − 1.35·35-s − 0.986·37-s − 0.320·39-s − 0.312·41-s + 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s + 0.269·55-s − 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.875675328772527825961545316041, −7.965954059246732200439764911687, −6.70418908817771650365413799438, −6.33644802733938487268952260308, −5.84871034939782789631443011618, −4.70993425864659877425168996437, −3.73471833099679066824930451074, −2.75044853344397076993328420584, −1.54953448111991999222312546648, 0,
1.54953448111991999222312546648, 2.75044853344397076993328420584, 3.73471833099679066824930451074, 4.70993425864659877425168996437, 5.84871034939782789631443011618, 6.33644802733938487268952260308, 6.70418908817771650365413799438, 7.965954059246732200439764911687, 8.875675328772527825961545316041
