| L(s) = 1 | − 3·3-s − 19.6·7-s + 9·9-s − 7.89·11-s − 74.5·13-s − 0.529·17-s − 89.7·19-s + 58.8·21-s + 174.·23-s − 27·27-s − 275.·29-s − 267.·31-s + 23.6·33-s + 11.3·37-s + 223.·39-s − 384.·41-s − 28.5·43-s + 289.·47-s + 41.8·49-s + 1.58·51-s + 256.·53-s + 269.·57-s − 174.·59-s − 732.·61-s − 176.·63-s − 498.·67-s − 524.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.05·7-s + 0.333·9-s − 0.216·11-s − 1.59·13-s − 0.00755·17-s − 1.08·19-s + 0.611·21-s + 1.58·23-s − 0.192·27-s − 1.76·29-s − 1.54·31-s + 0.125·33-s + 0.0505·37-s + 0.918·39-s − 1.46·41-s − 0.101·43-s + 0.898·47-s + 0.122·49-s + 0.00436·51-s + 0.664·53-s + 0.626·57-s − 0.384·59-s − 1.53·61-s − 0.353·63-s − 0.908·67-s − 0.914·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2874823176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2874823176\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 19.6T + 343T^{2} \) |
| 11 | \( 1 + 7.89T + 1.33e3T^{2} \) |
| 13 | \( 1 + 74.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.529T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 28.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 289.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 256.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 732.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 498.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 763.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 927.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 217.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 21.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 191.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 885.T + 9.12e5T^{2} \) |
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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
−8.869251372683342880174696392301, −7.47998583084770579207703852018, −7.15902113616109078784816594036, −6.30267635591806268666922163712, −5.45675416166174271728726119809, −4.78875369269426181060083124239, −3.74651286949643001999473561383, −2.82116969375403248776861386788, −1.79739884003427747230468093176, −0.23244026097565686757159438179,
0.23244026097565686757159438179, 1.79739884003427747230468093176, 2.82116969375403248776861386788, 3.74651286949643001999473561383, 4.78875369269426181060083124239, 5.45675416166174271728726119809, 6.30267635591806268666922163712, 7.15902113616109078784816594036, 7.47998583084770579207703852018, 8.869251372683342880174696392301
