| L(s) = 1 | + (1.86 + 1.77i)2-s + (0.219 + 4.61i)4-s + (−0.754 + 1.65i)5-s + (−0.561 + 2.31i)7-s + (−4.40 + 5.08i)8-s + (−4.33 + 1.73i)10-s + (−0.558 − 0.784i)11-s + (2.04 − 5.92i)13-s + (−5.15 + 3.31i)14-s + (−8.04 + 0.768i)16-s + (−0.0160 + 0.336i)17-s + (−0.0302 − 0.124i)19-s + (−7.78 − 3.11i)20-s + (0.352 − 2.45i)22-s + (3.05 − 2.40i)23-s + ⋯ |
| L(s) = 1 | + (1.31 + 1.25i)2-s + (0.109 + 2.30i)4-s + (−0.337 + 0.739i)5-s + (−0.212 + 0.875i)7-s + (−1.55 + 1.79i)8-s + (−1.37 + 0.549i)10-s + (−0.168 − 0.236i)11-s + (0.568 − 1.64i)13-s + (−1.37 + 0.885i)14-s + (−2.01 + 0.192i)16-s + (−0.00389 + 0.0816i)17-s + (−0.00693 − 0.0285i)19-s + (−1.74 − 0.697i)20-s + (0.0751 − 0.522i)22-s + (0.636 − 0.500i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.307013 + 2.59033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.307013 + 2.59033i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 + (2.01 + 7.93i)T \) |
| good | 2 | \( 1 + (-1.86 - 1.77i)T + (0.0951 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.754 - 1.65i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (0.561 - 2.31i)T + (-6.22 - 3.20i)T^{2} \) |
| 11 | \( 1 + (0.558 + 0.784i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-2.04 + 5.92i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (0.0160 - 0.336i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (0.0302 + 0.124i)T + (-16.8 + 8.70i)T^{2} \) |
| 23 | \( 1 + (-3.05 + 2.40i)T + (5.42 - 22.3i)T^{2} \) |
| 29 | \( 1 + (3.15 - 5.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.950 - 2.74i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (-2.91 - 5.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.28 - 2.72i)T + (23.7 - 33.3i)T^{2} \) |
| 43 | \( 1 + (-4.79 - 3.08i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (5.05 + 2.02i)T + (34.0 + 32.4i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 7.27i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.96 - 3.41i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.218 + 0.307i)T + (-19.9 - 57.6i)T^{2} \) |
| 71 | \( 1 + (0.388 + 8.14i)T + (-70.6 + 6.74i)T^{2} \) |
| 73 | \( 1 + (-4.48 + 6.29i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (8.90 - 1.71i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-7.61 + 0.726i)T + (81.5 - 15.7i)T^{2} \) |
| 89 | \( 1 + (-1.67 + 11.6i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.10 - 3.64i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.26821553830510339347073557428, −10.38852835091344259096629719958, −8.841994945519177213411861880923, −8.128472317194463332143024911366, −7.26367165016790371338397743830, −6.39889268804741935706683643637, −5.64296145823715933796009741798, −4.87523441353728054531939975185, −3.38466035680109013131146760259, −2.96269604387070004267714354420,
1.02920267126044303062262091477, 2.31244548604470769802628424975, 3.88710763707655495666792372330, 4.19058044785781323048647721032, 5.21930210766124628072676256306, 6.32995950201838043843337074676, 7.38038033814497510454266208481, 8.853790123852036904489417213184, 9.669720946741839102266754108368, 10.56845369372191410534346573825
