| L(s) = 1 | + (−0.281 + 0.959i)2-s + (−0.0697 − 0.0604i)3-s + (−0.841 − 0.540i)4-s + (−0.948 − 2.02i)5-s + (0.0776 − 0.0499i)6-s + (1.04 − 0.475i)7-s + (0.755 − 0.654i)8-s + (−0.425 − 2.96i)9-s + (2.21 − 0.339i)10-s + (3.69 − 1.08i)11-s + (0.0260 + 0.0886i)12-s + (1.25 + 0.571i)13-s + (0.162 + 1.13i)14-s + (−0.0563 + 0.198i)15-s + (0.415 + 0.909i)16-s + (−0.463 − 0.720i)17-s + ⋯ |
| L(s) = 1 | + (−0.199 + 0.678i)2-s + (−0.0402 − 0.0349i)3-s + (−0.420 − 0.270i)4-s + (−0.424 − 0.905i)5-s + (0.0317 − 0.0203i)6-s + (0.393 − 0.179i)7-s + (0.267 − 0.231i)8-s + (−0.141 − 0.987i)9-s + (0.698 − 0.107i)10-s + (1.11 − 0.327i)11-s + (0.00751 + 0.0255i)12-s + (0.346 + 0.158i)13-s + (0.0435 + 0.302i)14-s + (−0.0145 + 0.0513i)15-s + (0.103 + 0.227i)16-s + (−0.112 − 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.994995 - 0.224395i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.994995 - 0.224395i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.281 - 0.959i)T \) |
| 5 | \( 1 + (0.948 + 2.02i)T \) |
| 23 | \( 1 + (-0.229 + 4.79i)T \) |
| good | 3 | \( 1 + (0.0697 + 0.0604i)T + (0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.04 + 0.475i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.69 + 1.08i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 0.571i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.463 + 0.720i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.778 + 0.500i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.00469 + 0.00301i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.36 - 5.04i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 0.153i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.702 - 4.88i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (5.24 + 4.54i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 7.27iT - 47T^{2} \) |
| 53 | \( 1 + (-0.417 + 0.190i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.895 + 1.95i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.33 - 7.30i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.18 - 7.42i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.45 - 1.01i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (7.64 - 11.8i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.879 - 1.92i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 1.46i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (4.67 - 5.39i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-19.0 - 2.73i)T + (93.0 + 27.3i)T^{2} \) |
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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
−12.08041567962809512489480503017, −11.39329293750408796026296560807, −9.944249926613605384986907097592, −8.810153696621469804423200561542, −8.480736746832486784602889213639, −7.02902329967776255446719944532, −6.14550904237592394538723947347, −4.78911827485337292581013691397, −3.76849410432037319178504013477, −1.02308278683945208608112464870,
1.96691833332836969688956229680, 3.44216107694410968462113789677, 4.63705547455001636079046910197, 6.18071567555820015679087033808, 7.48771560633743559198052972301, 8.327508529027022009789497949960, 9.554514965699543272068051956009, 10.53495822450192252427176934012, 11.35747198565814517913386645738, 11.86682893487158111688546639731
