| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (1.68 + 1.47i)5-s + (−0.142 + 0.989i)6-s + (−0.663 + 1.21i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (1.78 + 1.34i)10-s + (3.43 − 2.97i)11-s + (−0.0713 + 0.997i)12-s + (−0.104 + 0.0572i)13-s + (−0.574 + 1.25i)14-s + (−1.79 + 1.33i)15-s + (0.959 − 0.281i)16-s + (1.93 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.752 + 0.658i)5-s + (−0.0580 + 0.404i)6-s + (−0.250 + 0.459i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.563 + 0.426i)10-s + (1.03 − 0.897i)11-s + (−0.0205 + 0.287i)12-s + (−0.0290 + 0.0158i)13-s + (−0.153 + 0.336i)14-s + (−0.464 + 0.343i)15-s + (0.239 − 0.0704i)16-s + (0.469 + 0.626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.21679 + 1.19393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21679 + 1.19393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 3 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (-1.68 - 1.47i)T \) |
| 23 | \( 1 + (1.83 - 4.43i)T \) |
| good | 7 | \( 1 + (0.663 - 1.21i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-3.43 + 2.97i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.104 - 0.0572i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.93 - 2.58i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.165 - 1.15i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (2.45 + 0.352i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (5.49 + 3.52i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.269 - 0.721i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.09 + 2.40i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.74 - 1.03i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (3.35 + 3.35i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.35 + 2.37i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.267 + 0.912i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.89 + 10.7i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (1.03 + 14.4i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-5.17 + 5.96i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-8.91 - 6.67i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (11.2 + 3.31i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.37 + 0.511i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (7.15 - 4.59i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-13.1 - 4.90i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.79400346961184327832936884640, −9.722239026616836228716413649819, −9.212892800757262826999845420671, −7.952958890678554256243595879166, −6.69232108173561172691029721453, −5.94144618412023754092998827991, −5.41069369020229150301246364373, −3.87902631897853939037467926462, −3.24539804989017851944473435511, −1.84667245430709476766578690181,
1.24086306260762318249311403524, 2.43060515644929048547123506304, 3.93239484446795256979501143532, 4.89597590096821378712279740006, 5.83200775342705197250383965350, 6.74928897204397435535324239630, 7.36853024027919708497856408312, 8.631505385902582044822736663687, 9.523938981575480435360783921849, 10.31048474703671025643108718239
