| L(s) = 1 | + (0.142 + 0.989i)2-s + (0.817 − 1.78i)3-s + (−0.959 + 0.281i)4-s + (−0.357 + 0.412i)5-s + (1.88 + 0.554i)6-s + (−3.96 + 2.54i)7-s + (−0.415 − 0.909i)8-s + (−0.569 − 0.657i)9-s + (−0.459 − 0.295i)10-s + (0.698 − 4.86i)11-s + (−0.279 + 1.94i)12-s + (2.05 + 1.31i)13-s + (−3.08 − 3.55i)14-s + (0.446 + 0.977i)15-s + (0.841 − 0.540i)16-s + (−1.58 − 0.466i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.471 − 1.03i)3-s + (−0.479 + 0.140i)4-s + (−0.159 + 0.184i)5-s + (0.770 + 0.226i)6-s + (−1.49 + 0.961i)7-s + (−0.146 − 0.321i)8-s + (−0.189 − 0.219i)9-s + (−0.145 − 0.0933i)10-s + (0.210 − 1.46i)11-s + (−0.0808 + 0.562i)12-s + (0.568 + 0.365i)13-s + (−0.823 − 0.950i)14-s + (0.115 + 0.252i)15-s + (0.210 − 0.135i)16-s + (−0.385 − 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.832067 + 0.113090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.832067 + 0.113090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-4.79 + 0.102i)T \) |
| good | 3 | \( 1 + (-0.817 + 1.78i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (0.357 - 0.412i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (3.96 - 2.54i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.698 + 4.86i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.05 - 1.31i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.58 + 0.466i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.11 - 0.328i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (2.00 + 0.588i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.66 + 5.84i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 2.11i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.96 + 2.26i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (4.45 - 9.76i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + (-2.93 + 1.88i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.76 - 1.77i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.18 + 4.77i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.0899 - 0.625i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.876 - 6.09i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (10.6 - 3.13i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.05 - 1.96i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (3.19 + 3.68i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (0.310 - 0.680i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.07 + 3.54i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−15.91671253935038760504891596470, −14.72497868856811062883550497067, −13.29122078151900095317916275008, −13.02834066675904881773607507402, −11.41408513063867576257192755254, −9.321279702013312493231459867254, −8.372563666679062504518311886888, −6.85735763705587468664019941409, −5.96898462781021610971408041398, −3.17671238151809933240179534843,
3.40300621886023635503152189152, 4.48916111198412176753861727355, 6.87860324482201103884954717200, 8.967993536883682982446265940172, 9.918545076615058601211154368790, 10.62090460164749608082435452604, 12.46902159884688375785334962139, 13.27377074796441204319587587045, 14.70350165616286616837911616480, 15.70429516158789819581739592879
