L(s) = 1 | + 8.86·3-s − 16.8i·5-s − 10.8i·7-s + 51.5·9-s + 35.1i·11-s + (−20.1 − 42.3i)13-s − 149. i·15-s − 30.3·17-s − 28.3i·19-s − 96.3i·21-s − 24.6·23-s − 159.·25-s + 218.·27-s + 290.·29-s + 219. i·31-s + ⋯ |
L(s) = 1 | + 1.70·3-s − 1.50i·5-s − 0.586i·7-s + 1.91·9-s + 0.964i·11-s + (−0.429 − 0.903i)13-s − 2.57i·15-s − 0.432·17-s − 0.342i·19-s − 1.00i·21-s − 0.223·23-s − 1.27·25-s + 1.55·27-s + 1.85·29-s + 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.54804 - 1.60956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.54804 - 1.60956i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (20.1 + 42.3i)T \) |
good | 3 | \( 1 - 8.86T + 27T^{2} \) |
| 5 | \( 1 + 16.8iT - 125T^{2} \) |
| 7 | \( 1 + 10.8iT - 343T^{2} \) |
| 11 | \( 1 - 35.1iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 30.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 24.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 290.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 83.6iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 293.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 166. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 76.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 184. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 197.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 321. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 368. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 843. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 690. iT - 9.12e5T^{2} \) |
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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
−12.28166378181333449004373345632, −10.36873810749932924517617637631, −9.563575750931252309748621213673, −8.683055014830795099706920841659, −8.034156911395797107479479920973, −7.03501727812601597695034663895, −4.96363584088814442342408417139, −4.13664624153263697809959941076, −2.63493039695462363420407603915, −1.16693116527910011872218892279,
2.25104309430081850129580706706, 2.95429020855017453737444134674, 4.08899427720723305158506816416, 6.18386359963952731515657926244, 7.18801518069643516720809942331, 8.177329205080405738672167929023, 9.064363722846416874891031294860, 9.980691910547106519053556663243, 11.01134945127010249505687596909, 12.11628772427321896485564612129