| L(s) = 1 | + 3-s − 2·5-s + 9-s + 2·13-s − 2·15-s + 6·17-s + 4·23-s − 25-s + 27-s − 2·29-s − 10·37-s + 2·39-s + 6·41-s + 8·43-s − 2·45-s + 4·47-s + 6·51-s − 6·53-s + 12·59-s + 2·61-s − 4·65-s + 4·67-s + 4·69-s + 12·71-s − 14·73-s − 75-s − 16·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.64·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.298·45-s + 0.583·47-s + 0.840·51-s − 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.481·69-s + 1.42·71-s − 1.63·73-s − 0.115·75-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.71037928752293, −13.10601840510854, −12.58616896809385, −12.34991152619512, −11.70127192049017, −11.23454293577045, −10.87757044105396, −10.12894351018903, −9.885509074002480, −9.149924139214570, −8.750552135289710, −8.270949785175619, −7.805490863957404, −7.267658499718556, −7.050439188254193, −6.207678959667091, −5.574614666057029, −5.243069773983696, −4.357289079390731, −3.950554255243510, −3.481775405500318, −2.967751511064909, −2.335148503747009, −1.430940291998748, −0.9598556435787038, 0,
0.9598556435787038, 1.430940291998748, 2.335148503747009, 2.967751511064909, 3.481775405500318, 3.950554255243510, 4.357289079390731, 5.243069773983696, 5.574614666057029, 6.207678959667091, 7.050439188254193, 7.267658499718556, 7.805490863957404, 8.270949785175619, 8.750552135289710, 9.149924139214570, 9.885509074002480, 10.12894351018903, 10.87757044105396, 11.23454293577045, 11.70127192049017, 12.34991152619512, 12.58616896809385, 13.10601840510854, 13.71037928752293
