| L(s) = 1 | + (−0.978 + 2.83i)3-s + (2.71 + 4.69i)7-s + (−7.08 − 5.54i)9-s + (−8.81 + 5.09i)11-s + (2.55 − 4.42i)13-s − 17.4i·17-s − 17.4·19-s + (−15.9 + 3.09i)21-s + (−16.3 − 9.41i)23-s + (22.6 − 14.6i)27-s + (29.0 − 16.7i)29-s + (−25.4 + 44.1i)31-s + (−5.81 − 29.9i)33-s − 0.605·37-s + (10.0 + 11.5i)39-s + ⋯ |
| L(s) = 1 | + (−0.326 + 0.945i)3-s + (0.387 + 0.671i)7-s + (−0.787 − 0.616i)9-s + (−0.801 + 0.462i)11-s + (0.196 − 0.340i)13-s − 1.02i·17-s − 0.920·19-s + (−0.760 + 0.147i)21-s + (−0.709 − 0.409i)23-s + (0.839 − 0.543i)27-s + (1.00 − 0.578i)29-s + (−0.822 + 1.42i)31-s + (−0.176 − 0.908i)33-s − 0.0163·37-s + (0.257 + 0.296i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0338 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0338 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4243652222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4243652222\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.978 - 2.83i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-2.71 - 4.69i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (8.81 - 5.09i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-2.55 + 4.42i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 17.4iT - 289T^{2} \) |
| 19 | \( 1 + 17.4T + 361T^{2} \) |
| 23 | \( 1 + (16.3 + 9.41i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-29.0 + 16.7i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (25.4 - 44.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 0.605T + 1.36e3T^{2} \) |
| 41 | \( 1 + (12.4 + 7.17i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (25.4 + 44.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.96 + 1.13i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 18.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (70.8 + 40.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.70 - 8.14i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (18.3 - 31.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 57.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 78.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (19.1 + 33.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-66.9 + 38.6i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 138. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (58.4 + 101. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.910539538559946676340231870204, −8.807044950984261908721582065364, −8.334849353517321281754884504827, −7.10128058165972667988047249221, −6.01899878609174565604741991274, −5.16343115477235358785114693630, −4.55333096052970659243556883893, −3.27835834592031466136834864100, −2.20468326872986413641267831960, −0.14571844318851328080005164984,
1.28659573571807007658762945034, 2.38480723155145808202763897639, 3.79475895657035491034736395483, 4.91775445817098964102439850894, 5.99740252029233723455202447976, 6.59426313374199894944373442320, 7.83111317043220760183587732794, 8.028325160134115606110258756804, 9.142446187019078133260491984149, 10.50759664831748553861748408337
