| L(s) = 1 | + (−1.35 − 1.86i)2-s + (−0.451 − 0.146i)3-s + (−1.02 + 3.16i)4-s + (2.19 − 0.420i)5-s + (0.338 + 1.04i)6-s + 3.03i·7-s + (2.92 − 0.951i)8-s + (−2.24 − 1.63i)9-s + (−3.76 − 3.53i)10-s + (−1.61 + 1.17i)11-s + (0.930 − 1.28i)12-s + (0.838 − 1.15i)13-s + (5.67 − 4.12i)14-s + (−1.05 − 0.132i)15-s + (−0.357 − 0.259i)16-s + (−1.76 + 0.574i)17-s + ⋯ |
| L(s) = 1 | + (−0.959 − 1.32i)2-s + (−0.260 − 0.0847i)3-s + (−0.514 + 1.58i)4-s + (0.982 − 0.187i)5-s + (0.138 + 0.425i)6-s + 1.14i·7-s + (1.03 − 0.336i)8-s + (−0.748 − 0.543i)9-s + (−1.19 − 1.11i)10-s + (−0.487 + 0.354i)11-s + (0.268 − 0.369i)12-s + (0.232 − 0.320i)13-s + (1.51 − 1.10i)14-s + (−0.272 − 0.0342i)15-s + (−0.0893 − 0.0649i)16-s + (−0.429 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.338102 - 0.279416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338102 - 0.279416i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.19 + 0.420i)T \) |
| good | 2 | \( 1 + (1.35 + 1.86i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.451 + 0.146i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 3.03iT - 7T^{2} \) |
| 11 | \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.838 + 1.15i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.76 - 0.574i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.279 + 0.859i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.95 + 2.69i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.22 - 3.76i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.99 + 6.12i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.24 + 3.09i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.48 - 1.07i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.59iT - 43T^{2} \) |
| 47 | \( 1 + (4.56 + 1.48i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.03 - 2.93i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.61 - 6.25i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (11.5 - 8.39i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 3.30i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.85 + 11.8i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.157 - 0.216i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.64 + 8.15i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (12.0 - 3.89i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.85 - 2.80i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (9.47 + 3.07i)T + (78.4 + 57.0i)T^{2} \) |
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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
−18.10654887689206407956242808015, −16.79318904475727817106591641484, −14.93013691025311420320926652276, −13.02837660570818258256266349106, −12.06627944208059647097832957114, −10.82063917633549326686182765412, −9.482819874120175830761398188808, −8.587410407659643003356611381504, −5.82385696664051564495606249014, −2.48425883173590507807753867406,
5.52198619498244300122709572032, 6.85744330449983430802479407420, 8.313327649596639150128601111549, 9.810108913165090789065643859250, 10.90587930528900373524074703507, 13.54259238772985619509116866189, 14.34254366238323078089611004746, 15.96599924038615834768568340323, 16.91057168293796945249074019103, 17.52117840616285466591855096200
