| L(s) = 1 | + 3-s − 5-s + 9-s − 13-s − 15-s − 6·17-s + 4·23-s + 25-s + 27-s − 10·29-s − 6·37-s − 39-s + 2·41-s + 4·43-s − 45-s − 7·49-s − 6·51-s − 6·53-s + 6·61-s + 65-s − 4·67-s + 4·69-s − 16·71-s − 2·73-s + 75-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.986·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 49-s − 0.840·51-s − 0.824·53-s + 0.768·61-s + 0.124·65-s − 0.488·67-s + 0.481·69-s − 1.89·71-s − 0.234·73-s + 0.115·75-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.416060489621200118497940918415, −7.48651432603210725392086041861, −7.05881776326155092462693764661, −6.13130217685388014086757536825, −5.09225276175166925177215351932, −4.33322549174996816277799124115, −3.53361530480953533803089286496, −2.61347722092406101275253466214, −1.63593081225043651097784490840, 0,
1.63593081225043651097784490840, 2.61347722092406101275253466214, 3.53361530480953533803089286496, 4.33322549174996816277799124115, 5.09225276175166925177215351932, 6.13130217685388014086757536825, 7.05881776326155092462693764661, 7.48651432603210725392086041861, 8.416060489621200118497940918415
