L(s) = 1 | + (0.707 − 0.707i)2-s + (1.36 + 0.564i)3-s − 1.00i·4-s + (0.382 − 0.923i)5-s + (1.36 − 0.564i)6-s + (1.35 + 3.27i)7-s + (−0.707 − 0.707i)8-s + (−0.581 − 0.581i)9-s + (−0.382 − 0.923i)10-s + (−4.35 + 1.80i)11-s + (0.564 − 1.36i)12-s − 5.47i·13-s + (3.27 + 1.35i)14-s + (1.04 − 1.04i)15-s − 1.00·16-s + (2.52 + 3.25i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.787 + 0.326i)3-s − 0.500i·4-s + (0.171 − 0.413i)5-s + (0.556 − 0.230i)6-s + (0.513 + 1.23i)7-s + (−0.250 − 0.250i)8-s + (−0.193 − 0.193i)9-s + (−0.121 − 0.292i)10-s + (−1.31 + 0.543i)11-s + (0.163 − 0.393i)12-s − 1.51i·13-s + (0.876 + 0.362i)14-s + (0.269 − 0.269i)15-s − 0.250·16-s + (0.613 + 0.789i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72813 - 0.414910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72813 - 0.414910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 + (-2.52 - 3.25i)T \) |
good | 3 | \( 1 + (-1.36 - 0.564i)T + (2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (-1.35 - 3.27i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (4.35 - 1.80i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + 5.47iT - 13T^{2} \) |
| 19 | \( 1 + (-0.857 + 0.857i)T - 19iT^{2} \) |
| 23 | \( 1 + (5.28 - 2.18i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (1.77 - 4.29i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-6.72 - 2.78i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (8.58 + 3.55i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.0547 - 0.132i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (0.587 + 0.587i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.85iT - 47T^{2} \) |
| 53 | \( 1 + (-6.54 + 6.54i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.33 + 3.33i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.76 - 11.4i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 + (-4.75 - 1.96i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-6.17 + 14.9i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-9.34 + 3.87i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.40 + 2.40i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.24iT - 89T^{2} \) |
| 97 | \( 1 + (-1.66 + 4.02i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.58333474375168429035590086074, −12.05237874406556887305949261434, −10.58944734047855909604616509939, −9.794036225432959731058348655474, −8.595257446648270767966260020264, −7.937539710473713299507249241118, −5.75362912733269015170707705360, −5.10940936921285466603210446442, −3.37017107446711173572010687131, −2.26074361973846656271127409516,
2.40532095765347409643954785236, 3.86255822775029665986653948268, 5.21445117041883226244285705797, 6.71607394649876030504279754078, 7.69476429607915333021088938358, 8.299384093439368943093016698727, 9.826410352954333617696921078013, 10.92717095718889013636888118660, 11.93340475843217307734455738669, 13.48527967756984319563869752967