| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 2·13-s + 15-s − 6·17-s + 2·21-s + 25-s − 27-s + 4·29-s − 8·31-s + 2·35-s + 6·37-s + 2·39-s + 6·43-s − 45-s − 8·47-s − 3·49-s + 6·51-s + 14·53-s + 4·59-s − 8·61-s − 2·63-s + 2·65-s − 12·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.320·39-s + 0.914·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.840·51-s + 1.92·53-s + 0.520·59-s − 1.02·61-s − 0.251·63-s + 0.248·65-s − 1.46·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.68753664235744, −13.12581465227012, −12.99459507070346, −12.35470099139536, −11.88941651537200, −11.46676722216927, −10.92683592723584, −10.52836117538919, −9.988368581260763, −9.447470715059513, −8.936211312632255, −8.580666096282500, −7.653334384644956, −7.437124523862783, −6.753521941899954, −6.403585143342340, −5.851366876231633, −5.234141989163469, −4.576329793891220, −4.240821784526796, −3.572659487570866, −2.883096531471483, −2.328456567714816, −1.560174174861016, −0.6014693745169226, 0,
0.6014693745169226, 1.560174174861016, 2.328456567714816, 2.883096531471483, 3.572659487570866, 4.240821784526796, 4.576329793891220, 5.234141989163469, 5.851366876231633, 6.403585143342340, 6.753521941899954, 7.437124523862783, 7.653334384644956, 8.580666096282500, 8.936211312632255, 9.447470715059513, 9.988368581260763, 10.52836117538919, 10.92683592723584, 11.46676722216927, 11.88941651537200, 12.35470099139536, 12.99459507070346, 13.12581465227012, 13.68753664235744
