| L(s) = 1 | + (0.951 + 0.309i)2-s + (−0.832 + 1.14i)3-s + (0.809 + 0.587i)4-s + (1.21 − 1.87i)5-s + (−1.14 + 0.832i)6-s + i·7-s + (0.587 + 0.809i)8-s + (0.306 + 0.943i)9-s + (1.73 − 1.40i)10-s + (−0.631 + 1.94i)11-s + (−1.34 + 0.437i)12-s + (5.71 − 1.85i)13-s + (−0.309 + 0.951i)14-s + (1.13 + 2.95i)15-s + (0.309 + 0.951i)16-s + (2.03 + 2.79i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.480 + 0.661i)3-s + (0.404 + 0.293i)4-s + (0.545 − 0.838i)5-s + (−0.468 + 0.340i)6-s + 0.377i·7-s + (0.207 + 0.286i)8-s + (0.102 + 0.314i)9-s + (0.549 − 0.444i)10-s + (−0.190 + 0.586i)11-s + (−0.389 + 0.126i)12-s + (1.58 − 0.515i)13-s + (−0.0825 + 0.254i)14-s + (0.292 + 0.764i)15-s + (0.0772 + 0.237i)16-s + (0.492 + 0.678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66118 + 0.889402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66118 + 0.889402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-1.21 + 1.87i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (0.832 - 1.14i)T + (-0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (0.631 - 1.94i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-5.71 + 1.85i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.03 - 2.79i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.47 - 1.79i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (8.18 + 2.65i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.86 - 4.98i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.30 + 5.30i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (9.12 - 2.96i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.56 + 4.82i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.31iT - 43T^{2} \) |
| 47 | \( 1 + (-2.10 + 2.89i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.47 + 3.40i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.11 - 3.43i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.36 + 7.28i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (8.44 + 11.6i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (1.21 + 0.886i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (12.8 + 4.18i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.472 - 0.343i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.19 - 3.01i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.727 - 2.23i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.50 + 7.58i)T + (-29.9 - 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.90153264278382794050893471671, −10.38269416081226518342396591017, −10.28615508384189166033582469110, −8.661534753631337072757203543397, −8.066032219782768233978919985935, −6.31965597757545392756453489264, −5.68268237623716924517781609901, −4.75519247681055419299466146060, −3.81144790040512250958639311681, −1.92108748702012609007987760390,
1.37538379949132933754917734927, 2.98699976155551529579535848962, 4.14261103505653667665522524115, 5.82216491518541214344658737899, 6.31864159385180397898014868079, 7.12210113302548692270753845760, 8.425936774930412306067202901624, 9.840968915382340703485294938144, 10.62863671695642703123157936975, 11.53739246085304733922216130781
