| L(s) = 1 | + 29.4·3-s + 49·7-s + 621.·9-s − 444.·11-s + 260.·13-s + 91.1·17-s − 450.·19-s + 1.44e3·21-s + 261.·23-s + 1.11e4·27-s + 6.07e3·29-s − 1.70e3·31-s − 1.30e4·33-s + 9.90e3·37-s + 7.65e3·39-s + 1.81e4·41-s + 2.07e4·43-s − 1.29e4·47-s + 2.40e3·49-s + 2.68e3·51-s − 3.39e4·53-s − 1.32e4·57-s + 3.77e4·59-s − 1.00e4·61-s + 3.04e4·63-s − 4.84e4·67-s + 7.67e3·69-s + ⋯ |
| L(s) = 1 | + 1.88·3-s + 0.377·7-s + 2.55·9-s − 1.10·11-s + 0.426·13-s + 0.0765·17-s − 0.286·19-s + 0.712·21-s + 0.102·23-s + 2.93·27-s + 1.34·29-s − 0.318·31-s − 2.09·33-s + 1.18·37-s + 0.805·39-s + 1.68·41-s + 1.70·43-s − 0.856·47-s + 0.142·49-s + 0.144·51-s − 1.66·53-s − 0.539·57-s + 1.41·59-s − 0.346·61-s + 0.966·63-s − 1.31·67-s + 0.194·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.247959300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.247959300\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 29.4T + 243T^{2} \) |
| 11 | \( 1 + 444.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 260.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 91.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 450.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 261.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.80e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.63e4T + 8.58e9T^{2} \) |
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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
−9.485148254806041099413341786267, −8.751448028556439091938098796528, −7.931186837604994151343428612174, −7.57851646626070265173714311412, −6.30118045225372026052087966216, −4.85617488300842359084010446900, −3.98270087575550623239194219338, −2.89225357810810253040449801388, −2.26435820543269356186847506857, −1.02976355427710040356717259164,
1.02976355427710040356717259164, 2.26435820543269356186847506857, 2.89225357810810253040449801388, 3.98270087575550623239194219338, 4.85617488300842359084010446900, 6.30118045225372026052087966216, 7.57851646626070265173714311412, 7.931186837604994151343428612174, 8.751448028556439091938098796528, 9.485148254806041099413341786267
