| L(s) = 1 | + (0.791 − 0.104i)2-s + (0.217 + 0.441i)3-s + (−1.31 + 0.352i)4-s + (−0.751 + 0.659i)5-s + (0.218 + 0.327i)6-s + (1.56 − 2.13i)7-s + (−2.48 + 1.02i)8-s + (1.67 − 2.18i)9-s + (−0.526 + 0.600i)10-s + (5.56 − 0.364i)11-s + (−0.442 − 0.504i)12-s + (1.76 + 1.76i)13-s + (1.01 − 1.85i)14-s + (−0.455 − 0.188i)15-s + (0.504 − 0.291i)16-s + (−3.85 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.559 − 0.0736i)2-s + (0.125 + 0.255i)3-s + (−0.658 + 0.176i)4-s + (−0.336 + 0.294i)5-s + (0.0892 + 0.133i)6-s + (0.591 − 0.806i)7-s + (−0.876 + 0.363i)8-s + (0.559 − 0.729i)9-s + (−0.166 + 0.189i)10-s + (1.67 − 0.110i)11-s + (−0.127 − 0.145i)12-s + (0.489 + 0.489i)13-s + (0.271 − 0.494i)14-s + (−0.117 − 0.0486i)15-s + (0.126 − 0.0728i)16-s + (−0.935 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86902 + 0.190742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86902 + 0.190742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.751 - 0.659i)T \) |
| 7 | \( 1 + (-1.56 + 2.13i)T \) |
| 17 | \( 1 + (3.85 - 1.45i)T \) |
| good | 2 | \( 1 + (-0.791 + 0.104i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (-0.217 - 0.441i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (-5.56 + 0.364i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 1.76i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.603 - 4.58i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-4.20 - 2.07i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.636 - 3.20i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-7.06 + 3.48i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.243 - 3.71i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-0.648 + 3.26i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (1.77 + 4.29i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.608 + 2.27i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.68 + 6.10i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (12.7 + 1.67i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (0.102 + 0.302i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-9.89 - 5.71i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.7 + 7.87i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (3.52 + 1.19i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (0.699 - 1.41i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (2.49 - 6.03i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-16.7 - 4.48i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (10.6 - 2.11i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.82518124632294269499895344421, −9.723475814853022799811353286797, −8.991626756093512246863768054818, −8.212976127003623800745374754196, −6.93424805493692041279976218877, −6.25888419583963864767096587154, −4.72030701740693727262227296316, −3.98390249700288679038653097199, −3.51573842901724620812640873122, −1.29684671527472708411945251036,
1.23929047431419941976819238100, 2.90121429571335261557122170640, 4.46209803953180229724109169312, 4.71452233046919338625395504749, 6.02799986724183066376764920376, 6.91765938398473592329178341401, 8.189152624214309547571955788452, 8.901063557886224679335403641293, 9.469745860947718236930237792130, 10.85047949368311711496318764304
