| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.943 + 1.45i)3-s + (−0.959 + 0.281i)4-s + (−0.0269 + 2.23i)5-s + (1.57 + 0.727i)6-s + (−2.74 + 1.76i)7-s + (0.415 + 0.909i)8-s + (−1.21 − 2.74i)9-s + (2.21 − 0.291i)10-s + (−0.103 + 0.719i)11-s + (0.496 − 1.65i)12-s + (−0.797 + 1.24i)13-s + (2.13 + 2.46i)14-s + (−3.22 − 2.14i)15-s + (0.841 − 0.540i)16-s + (0.487 − 1.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.544 + 0.838i)3-s + (−0.479 + 0.140i)4-s + (−0.0120 + 0.999i)5-s + (0.641 + 0.296i)6-s + (−1.03 + 0.667i)7-s + (0.146 + 0.321i)8-s + (−0.406 − 0.913i)9-s + (0.701 − 0.0921i)10-s + (−0.0312 + 0.217i)11-s + (0.143 − 0.479i)12-s + (−0.221 + 0.344i)13-s + (0.571 + 0.659i)14-s + (−0.831 − 0.554i)15-s + (0.210 − 0.135i)16-s + (0.118 − 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0175911 - 0.0897491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0175911 - 0.0897491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.943 - 1.45i)T \) |
| 5 | \( 1 + (0.0269 - 2.23i)T \) |
| 23 | \( 1 + (-1.82 - 4.43i)T \) |
| good | 7 | \( 1 + (2.74 - 1.76i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.103 - 0.719i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (0.797 - 1.24i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.487 + 1.66i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.66 + 5.65i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.91 + 6.52i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.61 + 3.54i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (7.28 + 8.40i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.240 - 0.208i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.85 - 8.43i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 0.296T + 47T^{2} \) |
| 53 | \( 1 + (1.32 + 2.06i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (4.64 - 7.22i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.22 + 1.92i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.0578 + 0.402i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (7.98 - 1.14i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.808 + 2.75i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.78 - 2.78i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (11.9 - 10.3i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.56 - 7.80i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.74 + 10.0i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.05160532014636822259507571742, −9.993247661255778083481064334848, −9.583540656960114254215587713731, −8.843302566892764977850363748364, −7.31637947442539581914181650433, −6.41235094146423458926045715424, −5.52686950226132429444665271552, −4.34335891352537429511998240214, −3.29399557745494082128666167470, −2.49634544923605291333242053005,
0.05456830790927125775142519703, 1.42881302034568807865751158830, 3.43552052944343397782537171442, 4.75863198167425802698043470377, 5.63732056275654823890035152182, 6.49718989399943745110138355336, 7.19813162012406719310900043949, 8.239478009681929639264166986325, 8.771014427717758362779810937534, 10.09733903326724048661849667189
