| L(s) = 1 | − 7.84·3-s − 107.·5-s + 25.5·7-s − 181.·9-s − 512.·11-s + 862.·13-s + 840.·15-s − 1.52e3·17-s + 1.54e3·19-s − 200.·21-s + 3.12e3·23-s + 8.34e3·25-s + 3.32e3·27-s − 947.·29-s − 339.·31-s + 4.01e3·33-s − 2.73e3·35-s − 7.44e3·37-s − 6.76e3·39-s + 5.11e3·41-s + 1.84e3·43-s + 1.94e4·45-s + 1.71e4·47-s − 1.61e4·49-s + 1.19e4·51-s + 1.80e4·53-s + 5.48e4·55-s + ⋯ |
| L(s) = 1 | − 0.503·3-s − 1.91·5-s + 0.196·7-s − 0.746·9-s − 1.27·11-s + 1.41·13-s + 0.963·15-s − 1.27·17-s + 0.980·19-s − 0.0990·21-s + 1.23·23-s + 2.67·25-s + 0.878·27-s − 0.209·29-s − 0.0634·31-s + 0.642·33-s − 0.377·35-s − 0.894·37-s − 0.712·39-s + 0.475·41-s + 0.152·43-s + 1.43·45-s + 1.13·47-s − 0.961·49-s + 0.642·51-s + 0.884·53-s + 2.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 3 | \( 1 + 7.84T + 243T^{2} \) |
| 5 | \( 1 + 107.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 25.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 512.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 862.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.52e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 947.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 339.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.44e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.11e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.70e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.79e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.96e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.59e5T + 8.58e9T^{2} \) |
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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
−8.898618688790946536442671182915, −8.414444316726508383727905578372, −7.57828115914648979526830748265, −6.76806032979249771746151687197, −5.51863526803212538435450055627, −4.70709580976281467539195174143, −3.66070991376605927114086365434, −2.80200430752710043782088438027, −0.874146147763291120491155032018, 0,
0.874146147763291120491155032018, 2.80200430752710043782088438027, 3.66070991376605927114086365434, 4.70709580976281467539195174143, 5.51863526803212538435450055627, 6.76806032979249771746151687197, 7.57828115914648979526830748265, 8.414444316726508383727905578372, 8.898618688790946536442671182915
