| L(s) = 1 | − 9·3-s + 60.6·5-s − 130.·7-s + 81·9-s − 121·11-s + 111.·13-s − 546.·15-s + 1.26e3·17-s − 469.·19-s + 1.17e3·21-s − 115.·23-s + 556.·25-s − 729·27-s − 4.40e3·29-s + 2.78e3·31-s + 1.08e3·33-s − 7.89e3·35-s + 6.32e3·37-s − 1.00e3·39-s − 1.02e4·41-s + 1.62e3·43-s + 4.91e3·45-s − 6.53e3·47-s + 143.·49-s − 1.13e4·51-s + 2.69e4·53-s − 7.34e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.08·5-s − 1.00·7-s + 0.333·9-s − 0.301·11-s + 0.182·13-s − 0.626·15-s + 1.05·17-s − 0.298·19-s + 0.579·21-s − 0.0455·23-s + 0.177·25-s − 0.192·27-s − 0.971·29-s + 0.520·31-s + 0.174·33-s − 1.08·35-s + 0.759·37-s − 0.105·39-s − 0.954·41-s + 0.134·43-s + 0.361·45-s − 0.431·47-s + 0.00853·49-s − 0.611·51-s + 1.31·53-s − 0.327·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 11 | \( 1 + 121T \) |
| good | 5 | \( 1 - 60.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 130.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 111.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 469.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 115.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.32e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.02e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.12e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.90e4T + 8.58e9T^{2} \) |
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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
−9.918057345269114157216869164392, −8.958048813786482018598917505916, −7.70228470301141485836561407620, −6.61105327183669637135644410685, −5.92954522753762227670602578101, −5.20747558755981560172522915603, −3.78412875507865944775225287702, −2.59875674486608170771005530978, −1.33360695352125014476935743814, 0,
1.33360695352125014476935743814, 2.59875674486608170771005530978, 3.78412875507865944775225287702, 5.20747558755981560172522915603, 5.92954522753762227670602578101, 6.61105327183669637135644410685, 7.70228470301141485836561407620, 8.958048813786482018598917505916, 9.918057345269114157216869164392
