| L(s) = 1 | + 8.76·3-s + 49·7-s − 166.·9-s + 395.·11-s + 1.00e3·13-s − 430.·17-s + 2.67e3·19-s + 429.·21-s − 1.06e3·23-s − 3.58e3·27-s + 2.11e3·29-s − 3.32e3·31-s + 3.46e3·33-s − 2.81e3·37-s + 8.81e3·39-s − 1.48e4·41-s − 1.40e3·43-s + 2.55e3·47-s + 2.40e3·49-s − 3.77e3·51-s + 1.45e4·53-s + 2.34e4·57-s + 3.44e4·59-s + 1.97e4·61-s − 8.13e3·63-s − 5.13e4·67-s − 9.34e3·69-s + ⋯ |
| L(s) = 1 | + 0.562·3-s + 0.377·7-s − 0.683·9-s + 0.984·11-s + 1.65·13-s − 0.360·17-s + 1.69·19-s + 0.212·21-s − 0.420·23-s − 0.946·27-s + 0.466·29-s − 0.620·31-s + 0.553·33-s − 0.338·37-s + 0.928·39-s − 1.37·41-s − 0.115·43-s + 0.168·47-s + 0.142·49-s − 0.203·51-s + 0.713·53-s + 0.954·57-s + 1.28·59-s + 0.678·61-s − 0.258·63-s − 1.39·67-s − 0.236·69-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
| \(L(3)\) |
\(\approx\) |
\(3.301288541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.301288541\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 8.76T + 243T^{2} \) |
| 11 | \( 1 - 395.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.00e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 430.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.67e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.48e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.55e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.39e5T + 8.58e9T^{2} \) |
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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
−9.465294965109998132873203336415, −8.746629353708337195625086680857, −8.197695443072541614844316926985, −7.10002815371140411806252021023, −6.12122225829103739177959221269, −5.24979374099709880521296697988, −3.87482331299647477795089590923, −3.25929217299564859809938768826, −1.88405882313619701159436213026, −0.871513315998929025989865794580,
0.871513315998929025989865794580, 1.88405882313619701159436213026, 3.25929217299564859809938768826, 3.87482331299647477795089590923, 5.24979374099709880521296697988, 6.12122225829103739177959221269, 7.10002815371140411806252021023, 8.197695443072541614844316926985, 8.746629353708337195625086680857, 9.465294965109998132873203336415
