L(s) = 1 | + (−0.147 − 0.0222i)5-s + (0.623 − 0.781i)7-s + (−0.733 + 0.680i)9-s + (0.326 + 0.302i)11-s + (0.123 − 1.64i)17-s + (−0.5 − 0.866i)19-s + (−0.109 − 1.46i)23-s + (−0.934 − 0.288i)25-s + (−0.109 + 0.101i)35-s + (1.03 + 1.29i)43-s + (0.123 − 0.0841i)45-s + (1.19 − 0.367i)47-s + (−0.222 − 0.974i)49-s + (−0.0414 − 0.0520i)55-s + (−0.367 − 0.250i)61-s + ⋯ |
L(s) = 1 | + (−0.147 − 0.0222i)5-s + (0.623 − 0.781i)7-s + (−0.733 + 0.680i)9-s + (0.326 + 0.302i)11-s + (0.123 − 1.64i)17-s + (−0.5 − 0.866i)19-s + (−0.109 − 1.46i)23-s + (−0.934 − 0.288i)25-s + (−0.109 + 0.101i)35-s + (1.03 + 1.29i)43-s + (0.123 − 0.0841i)45-s + (1.19 − 0.367i)47-s + (−0.222 − 0.974i)49-s + (−0.0414 − 0.0520i)55-s + (−0.367 − 0.250i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.126062518\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126062518\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 5 | \( 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)T^{2} \) |
| 11 | \( 1 + (-0.326 - 0.302i)T + (0.0747 + 0.997i)T^{2} \) |
| 13 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.123 + 1.64i)T + (-0.988 - 0.149i)T^{2} \) |
| 23 | \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-1.19 + 0.367i)T + (0.826 - 0.563i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 61 | \( 1 + (0.367 + 0.250i)T + (0.365 + 0.930i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-1.91 - 0.589i)T + (0.826 + 0.563i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.459239684384883727223143686902, −7.80845883817131689070398440599, −7.17512544098289330007158446015, −6.44171415068684720625661390562, −5.41867639342924537154388930609, −4.66519150255053058755897358483, −4.16618188695783181813776576120, −2.86597756072453365318302850623, −2.15638512218014445912180281272, −0.66259167544785012370933669529,
1.43801208066120929083031849009, 2.34199442303453918256386676525, 3.61925532750308278196822847884, 3.96296262719699062837096441619, 5.34794458429710145997514328862, 5.85742446949397370730030462208, 6.36792598577234351728054798713, 7.61771245445280761989544830818, 8.107849811791018874723571056489, 8.861298174463837178002100996686