| L(s) = 1 | + 37·7-s + 19·13-s − 163·19-s − 125·25-s − 308·31-s − 323·37-s − 520·43-s + 1.02e3·49-s − 719·61-s − 127·67-s − 919·73-s + 1.38e3·79-s + 703·91-s − 523·97-s + 1.80e3·103-s + 646·109-s + ⋯ |
| L(s) = 1 | + 1.99·7-s + 0.405·13-s − 1.96·19-s − 25-s − 1.78·31-s − 1.43·37-s − 1.84·43-s + 2.99·49-s − 1.50·61-s − 0.231·67-s − 1.47·73-s + 1.97·79-s + 0.809·91-s − 0.547·97-s + 1.72·103-s + 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 - 37 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 323 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 520 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 719 T + p^{3} T^{2} \) |
| 67 | \( 1 + 127 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 919 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1387 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 523 T + p^{3} T^{2} \) |
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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
−8.501741302406500381408280008355, −7.916700916556244319148364586658, −7.09822042336490327950938505073, −6.05636350427777623540752710123, −5.19457605164743382896622988428, −4.47392699962665856285863788563, −3.63860728891274218659461364561, −2.03787750194275909180600459162, −1.61627089543244390839691596148, 0,
1.61627089543244390839691596148, 2.03787750194275909180600459162, 3.63860728891274218659461364561, 4.47392699962665856285863788563, 5.19457605164743382896622988428, 6.05636350427777623540752710123, 7.09822042336490327950938505073, 7.916700916556244319148364586658, 8.501741302406500381408280008355
