| L(s) = 1 | − 8·4-s − 20·7-s + 64·16-s − 56·19-s − 125·25-s + 160·28-s − 308·31-s − 110·37-s − 520·43-s + 57·49-s + 182·61-s − 512·64-s + 880·67-s − 1.19e3·73-s + 448·76-s + 884·79-s + 1.33e3·97-s + 1.00e3·100-s + 1.82e3·103-s + 646·109-s − 1.28e3·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.07·7-s + 16-s − 0.676·19-s − 25-s + 1.07·28-s − 1.78·31-s − 0.488·37-s − 1.84·43-s + 0.166·49-s + 0.382·61-s − 64-s + 1.60·67-s − 1.90·73-s + 0.676·76-s + 1.25·79-s + 1.39·97-s + 100-s + 1.74·103-s + 0.567·109-s − 1.07·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5836889778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5836889778\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 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 | \( 1 - 182 T + p^{3} T^{2} \) |
| 67 | \( 1 - 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−9.128259920149454599105881726584, −8.474264327392467569383426992496, −7.55326526238465471383452984841, −6.60824334894450726365154115399, −5.78491532683739033575566513445, −4.94986190520765637234160228414, −3.88153215661751968936986651283, −3.31385040095635340929951606062, −1.88323948183718506439718852949, −0.36382943752534658890646139082,
0.36382943752534658890646139082, 1.88323948183718506439718852949, 3.31385040095635340929951606062, 3.88153215661751968936986651283, 4.94986190520765637234160228414, 5.78491532683739033575566513445, 6.60824334894450726365154115399, 7.55326526238465471383452984841, 8.474264327392467569383426992496, 9.128259920149454599105881726584
