L(s) = 1 | + (1.80 − 3.12i)2-s + (13.5 − 7.79i)3-s + (25.4 + 44.1i)4-s + (71.9 + 41.5i)5-s − 56.2i·6-s + (77.0 − 334. i)7-s + 414.·8-s + (121.5 − 210. i)9-s + (259. − 149. i)10-s + (221. + 383. i)11-s + (688. + 397. i)12-s − 696. i·13-s + (−904. − 843. i)14-s + 1.29e3·15-s + (−883. + 1.53e3i)16-s + (−5.44e3 + 3.14e3i)17-s + ⋯ |
L(s) = 1 | + (0.225 − 0.390i)2-s + (0.5 − 0.288i)3-s + (0.398 + 0.690i)4-s + (0.575 + 0.332i)5-s − 0.260i·6-s + (0.224 − 0.974i)7-s + 0.810·8-s + (0.166 − 0.288i)9-s + (0.259 − 0.149i)10-s + (0.166 + 0.287i)11-s + (0.398 + 0.230i)12-s − 0.317i·13-s + (−0.329 − 0.307i)14-s + 0.383·15-s + (−0.215 + 0.373i)16-s + (−1.10 + 0.640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.26303 - 0.404136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26303 - 0.404136i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-13.5 + 7.79i)T \) |
| 7 | \( 1 + (-77.0 + 334. i)T \) |
good | 2 | \( 1 + (-1.80 + 3.12i)T + (-32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (-71.9 - 41.5i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-221. - 383. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 696. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (5.44e3 - 3.14e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (2.32e3 + 1.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (7.88e3 - 1.36e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.32e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-4.11e4 + 2.37e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-5.07e3 + 8.79e3i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 3.81e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.51e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (4.35e4 + 2.51e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-9.97e4 - 1.72e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-3.35e5 + 1.93e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.02e4 + 5.89e3i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.92e5 - 3.32e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 1.56e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.25e5 + 1.87e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.65e4 - 2.87e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 9.84e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (2.27e5 + 1.31e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 5.75e5iT - 8.32e11T^{2} \) |
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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
−17.02220845672826497813508722912, −15.35585928249352104300795482125, −13.80404556805187924935494271377, −13.05380030179075276747864559540, −11.44994331964439743807276233176, −10.09953342764295974119559452149, −8.095403499820281522282737613740, −6.76106962101120126158108468666, −3.93885903225597496845093492019, −2.05946532982861359310637616495,
2.08587472078149138640685531628, 4.96454371995627302731878644212, 6.44405189219881082086790383539, 8.589419392436654947978968579292, 9.894488479715852686764606807461, 11.49985651740546567700959473670, 13.35723133526094397364076134727, 14.50446386057366380717892750769, 15.48215399207945731886409472959, 16.56270965691785323284418239638