| L(s) = 1 | + 62.9·5-s + 85.2·7-s + 69.5·11-s + 169·13-s + 1.37e3·17-s + 1.24e3·19-s − 749.·23-s + 843.·25-s + 2.08e3·29-s + 3.86e3·31-s + 5.36e3·35-s − 1.47e4·37-s + 1.03e4·41-s + 6.75e3·43-s + 6.80e3·47-s − 9.54e3·49-s + 2.99e4·53-s + 4.38e3·55-s + 2.03e3·59-s + 2.10e4·61-s + 1.06e4·65-s − 4.52e4·67-s + 6.02e4·71-s − 1.53e3·73-s + 5.93e3·77-s + 3.03e4·79-s − 1.86e4·83-s + ⋯ |
| L(s) = 1 | + 1.12·5-s + 0.657·7-s + 0.173·11-s + 0.277·13-s + 1.15·17-s + 0.788·19-s − 0.295·23-s + 0.269·25-s + 0.459·29-s + 0.722·31-s + 0.740·35-s − 1.76·37-s + 0.965·41-s + 0.557·43-s + 0.449·47-s − 0.567·49-s + 1.46·53-s + 0.195·55-s + 0.0759·59-s + 0.724·61-s + 0.312·65-s − 1.23·67-s + 1.41·71-s − 0.0337·73-s + 0.114·77-s + 0.546·79-s − 0.297·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.758800589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.758800589\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 5 | \( 1 - 62.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 85.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 69.5T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.03e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.99e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.03e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.53e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e5T + 8.58e9T^{2} \) |
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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
−9.434625981259235910506775844448, −8.512841268014264647326964440661, −7.68841601282581971335006398636, −6.70435151832609779043348568426, −5.70945043144329826073824604647, −5.21243462806217000946138861843, −3.98030128601919350695001518180, −2.81075824300413417956370900430, −1.73754176313836923965239682427, −0.922578918729037410351671615835,
0.922578918729037410351671615835, 1.73754176313836923965239682427, 2.81075824300413417956370900430, 3.98030128601919350695001518180, 5.21243462806217000946138861843, 5.70945043144329826073824604647, 6.70435151832609779043348568426, 7.68841601282581971335006398636, 8.512841268014264647326964440661, 9.434625981259235910506775844448
