L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (3.56 + 3.50i)5-s + (−0.315 − 0.182i)7-s − 2.82·8-s + (6.81 − 1.89i)10-s + (8.32 + 4.80i)11-s + (6.62 − 3.82i)13-s + (−0.446 + 0.257i)14-s + (−2.00 + 3.46i)16-s − 12.3·17-s + 5.36·19-s + (2.49 − 9.68i)20-s + (11.7 − 6.79i)22-s + (6.03 + 10.4i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.713 + 0.700i)5-s + (−0.0451 − 0.0260i)7-s − 0.353·8-s + (0.681 − 0.189i)10-s + (0.756 + 0.436i)11-s + (0.509 − 0.294i)13-s + (−0.0319 + 0.0184i)14-s + (−0.125 + 0.216i)16-s − 0.725·17-s + 0.282·19-s + (0.124 − 0.484i)20-s + (0.535 − 0.308i)22-s + (0.262 + 0.454i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.616654430\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.616654430\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.56 - 3.50i)T \) |
good | 7 | \( 1 + (0.315 + 0.182i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.32 - 4.80i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-6.62 + 3.82i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 19 | \( 1 - 5.36T + 361T^{2} \) |
| 23 | \( 1 + (-6.03 - 10.4i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-27.1 - 15.6i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (2.45 + 4.26i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-54.8 + 31.6i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-46.6 - 26.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (14.1 - 24.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 41.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-97.2 + 56.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.9 - 39.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (39.9 - 23.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (25.3 - 43.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (29.5 - 51.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 8.93iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (113. + 65.6i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.08085510296171886878486067372, −9.431408710751075305159096004198, −8.574260787933813701689605648414, −7.22094988430893336933616906913, −6.44961201768394407187608008897, −5.61450739112935962746663281288, −4.48968454315567156933154870262, −3.42106752050074162174425138088, −2.42035133222729139991563025922, −1.24248343382078872577502683684,
0.936698577244131893981809462022, 2.45363282374856671382226167955, 3.92737013805779661833108041662, 4.71935078801206758554997140213, 5.84259844036587686106404787897, 6.34953049161475277660249570267, 7.37889937410598570520304640185, 8.610419633729029410398461382543, 8.939049473501936446214058132811, 9.862049949095200171704144883411