| L(s) = 1 | + 9·3-s − 30.2·5-s − 238.·7-s + 81·9-s + 421.·11-s − 217.·13-s − 272.·15-s − 1.38e3·17-s − 2.66e3·19-s − 2.14e3·21-s − 529·23-s − 2.20e3·25-s + 729·27-s − 1.08e3·29-s + 1.84e3·31-s + 3.79e3·33-s + 7.21e3·35-s + 9.90e3·37-s − 1.95e3·39-s − 1.35e4·41-s − 1.37e3·43-s − 2.45e3·45-s − 1.55e3·47-s + 3.99e4·49-s − 1.25e4·51-s − 1.40e3·53-s − 1.27e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.541·5-s − 1.83·7-s + 0.333·9-s + 1.05·11-s − 0.356·13-s − 0.312·15-s − 1.16·17-s − 1.69·19-s − 1.06·21-s − 0.208·23-s − 0.706·25-s + 0.192·27-s − 0.239·29-s + 0.344·31-s + 0.606·33-s + 0.996·35-s + 1.18·37-s − 0.205·39-s − 1.25·41-s − 0.113·43-s − 0.180·45-s − 0.102·47-s + 2.37·49-s − 0.673·51-s − 0.0685·53-s − 0.569·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8337354837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8337354837\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 30.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 238.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 421.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 217.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.38e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.66e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 1.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.55e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.40e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.46e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.14e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.25e4T + 8.58e9T^{2} \) |
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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
−9.095843713831495762279718481768, −8.505807960951567608605933150753, −7.37590223115656047711936470346, −6.55936656998270860391233203171, −6.14350461042266734069242409052, −4.40973349926916767961873281021, −3.86986164446887615062289129318, −2.94998663909830863704050619218, −1.96159338571418387764590349338, −0.35673487103208124107068290706,
0.35673487103208124107068290706, 1.96159338571418387764590349338, 2.94998663909830863704050619218, 3.86986164446887615062289129318, 4.40973349926916767961873281021, 6.14350461042266734069242409052, 6.55936656998270860391233203171, 7.37590223115656047711936470346, 8.505807960951567608605933150753, 9.095843713831495762279718481768
