| L(s) = 1 | + (−1.93 − 2.45i)2-s + (1.30 + 1.13i)3-s + (−1.35 + 5.59i)4-s + (2.11 + 3.29i)5-s + (0.257 − 5.40i)6-s + (0.841 − 2.10i)7-s + (4.97 − 2.27i)8-s + (0.426 + 2.96i)9-s + (3.99 − 11.5i)10-s + (−19.8 + 0.945i)11-s + (−8.11 + 5.77i)12-s + (1.94 + 20.4i)13-s + (−6.78 + 1.99i)14-s + (−0.964 + 6.70i)15-s + (5.26 + 2.71i)16-s + (5.41 + 22.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.965 − 1.22i)2-s + (0.436 + 0.378i)3-s + (−0.339 + 1.39i)4-s + (0.423 + 0.658i)5-s + (0.0428 − 0.900i)6-s + (0.120 − 0.300i)7-s + (0.622 − 0.284i)8-s + (0.0474 + 0.329i)9-s + (0.399 − 1.15i)10-s + (−1.80 + 0.0859i)11-s + (−0.676 + 0.481i)12-s + (0.149 + 1.57i)13-s + (−0.484 + 0.142i)14-s + (−0.0643 + 0.447i)15-s + (0.328 + 0.169i)16-s + (0.318 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.732319 + 0.296138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.732319 + 0.296138i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.30 - 1.13i)T \) |
| 67 | \( 1 + (6.50 + 66.6i)T \) |
| good | 2 | \( 1 + (1.93 + 2.45i)T + (-0.943 + 3.88i)T^{2} \) |
| 5 | \( 1 + (-2.11 - 3.29i)T + (-10.3 + 22.7i)T^{2} \) |
| 7 | \( 1 + (-0.841 + 2.10i)T + (-35.4 - 33.8i)T^{2} \) |
| 11 | \( 1 + (19.8 - 0.945i)T + (120. - 11.5i)T^{2} \) |
| 13 | \( 1 + (-1.94 - 20.4i)T + (-165. + 31.9i)T^{2} \) |
| 17 | \( 1 + (-5.41 - 22.3i)T + (-256. + 132. i)T^{2} \) |
| 19 | \( 1 + (1.58 - 0.636i)T + (261. - 249. i)T^{2} \) |
| 23 | \( 1 + (22.2 + 4.29i)T + (491. + 196. i)T^{2} \) |
| 29 | \( 1 + (-10.2 + 17.7i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (1.92 - 20.1i)T + (-943. - 181. i)T^{2} \) |
| 37 | \( 1 + (-34.6 - 59.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-26.9 - 28.2i)T + (-79.9 + 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-5.99 + 20.4i)T + (-1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 + (0.810 + 2.34i)T + (-1.73e3 + 1.36e3i)T^{2} \) |
| 53 | \( 1 + (16.7 + 57.1i)T + (-2.36e3 + 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-16.8 - 36.7i)T + (-2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (83.0 + 3.95i)T + (3.70e3 + 353. i)T^{2} \) |
| 71 | \( 1 + (30.1 - 124. i)T + (-4.48e3 - 2.30e3i)T^{2} \) |
| 73 | \( 1 + (-4.05 + 85.1i)T + (-5.30e3 - 506. i)T^{2} \) |
| 79 | \( 1 + (-55.9 + 39.8i)T + (2.04e3 - 5.89e3i)T^{2} \) |
| 83 | \( 1 + (46.0 + 23.7i)T + (3.99e3 + 5.61e3i)T^{2} \) |
| 89 | \( 1 + (24.0 + 27.7i)T + (-1.12e3 + 7.84e3i)T^{2} \) |
| 97 | \( 1 + (14.6 - 8.47i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.05143965489317673765808330273, −10.94005377788659355168527727132, −10.33206798904843283914130670477, −9.776913388792627636562852239369, −8.545911413098330730674060630810, −7.79405341073670595255999285657, −6.18820494556550061463121202370, −4.33671961180891828351505309596, −2.90919112063027975681651848509, −1.91016283366416010720132403469,
0.56966693102184839368659263913, 2.74048343298181642058651917750, 5.27803895354384747467599277195, 5.81004509390823281163871480987, 7.52265012968246760350664755321, 7.86142428081636024481271615569, 8.870708467121705415399200398669, 9.723504314610466157969193793996, 10.70047675666581731871921939525, 12.43428280964444383413505037885
