L(s) = 1 | + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (0.0348 + 0.999i)11-s + (0.997 − 0.0697i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.882 + 0.469i)20-s + (−0.0348 + 0.999i)22-s + (−0.882 + 0.469i)23-s + ⋯ |
L(s) = 1 | + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (0.0348 + 0.999i)11-s + (0.997 − 0.0697i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.882 + 0.469i)20-s + (−0.0348 + 0.999i)22-s + (−0.882 + 0.469i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.254915807 + 0.5255627773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.254915807 + 0.5255627773i\) |
\(L(1)\) |
\(\approx\) |
\(2.188180270 + 0.2130045192i\) |
\(L(1)\) |
\(\approx\) |
\(2.188180270 + 0.2130045192i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.997 + 0.0697i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 7 | \( 1 + (-0.882 - 0.469i)T \) |
| 11 | \( 1 + (0.0348 + 0.999i)T \) |
| 13 | \( 1 + (0.997 - 0.0697i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (-0.882 + 0.469i)T \) |
| 29 | \( 1 + (-0.997 - 0.0697i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.241 - 0.970i)T \) |
| 43 | \( 1 + (-0.559 + 0.829i)T \) |
| 47 | \( 1 + (0.719 - 0.694i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.559 - 0.829i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.669 + 0.743i)T \) |
| 73 | \( 1 + (-0.669 - 0.743i)T \) |
| 79 | \( 1 + (0.374 - 0.927i)T \) |
| 83 | \( 1 + (-0.997 - 0.0697i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (0.0348 + 0.999i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.99562889522363262359699620421, −21.43038353759609594631807266523, −20.79234764492662243660026075446, −19.923631591520080429571828546433, −18.920322422146035746495222886061, −18.34523144098994176903675163403, −16.852257232388778351056793144221, −16.44326761099014765282646614578, −15.719371274694469874136710981875, −14.56849221798409397577201266042, −13.96654351799920478724835197452, −13.07755557573101368526271714453, −12.66086698100290926263857004798, −11.67458499947747285415139559258, −10.63712816963811173112816836249, −9.95356495472615545750530860858, −8.880437181802706746635628435030, −7.95575343635737537677461543656, −6.501743935676831276716036400416, −5.90096327921237718484045020411, −5.56402295952677560856710764957, −4.03207746250745898245695593509, −3.3523351642967928228349228255, −2.26752556238993435048376629398, −1.27012449779415783493445986471,
1.39114070589808314823963285108, 2.4755824621544474995886492433, 3.33066357276261369008954060828, 4.27922110418294639348163566623, 5.37939573193308480826320431161, 6.15599728232556893191948528430, 6.908374868266195752968026037942, 7.62068420597931437004096241317, 9.20285064836050788102856416649, 9.98220901458974320380701409988, 10.72062599300496559378039875481, 11.70594124388368740046162224255, 12.68137406519932863004693985931, 13.45055665404885983865653645611, 13.81010448898384714323663532446, 14.856916405694816272646824636837, 15.61756143753943605483638510925, 16.44807320860829909862599304626, 17.226792034007748979953002863627, 18.13877716269071459713148485525, 19.08162648701253977302265476205, 20.28149421368536705037749242870, 20.48579585095425668069135021302, 21.61567882869426801315890450911, 22.19224651955730165562200162857