L(s) = 1 | + (3.99 + 0.0498i)2-s + 5.19i·3-s + (15.9 + 0.399i)4-s + (−0.259 + 20.7i)6-s + 35.2i·7-s + (63.9 + 2.39i)8-s − 27·9-s − 3.95i·11-s + (−2.07 + 83.1i)12-s + 41.0·13-s + (−1.76 + 141. i)14-s + (255. + 12.7i)16-s + 186.·17-s + (−107. − 1.34i)18-s + 580. i·19-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0124i)2-s + 0.577i·3-s + (0.999 + 0.0249i)4-s + (−0.00720 + 0.577i)6-s + 0.720i·7-s + (0.999 + 0.0374i)8-s − 0.333·9-s − 0.0326i·11-s + (−0.0144 + 0.577i)12-s + 0.242·13-s + (−0.00898 + 0.720i)14-s + (0.998 + 0.0498i)16-s + 0.646·17-s + (−0.333 − 0.00415i)18-s + 1.60i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0249 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0249 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.783169858\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.783169858\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.99 - 0.0498i)T \) |
| 3 | \( 1 - 5.19iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 35.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 3.95iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 41.0T + 2.85e4T^{2} \) |
| 17 | \( 1 - 186.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 580. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 472. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 979.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 33.2iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 623.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 154.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.67e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.92e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.78e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.48e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 488.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.49e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.62e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.95e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.75e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 4.86e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 513.T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.21e4T + 8.85e7T^{2} \) |
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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
−11.56901568446361105278369517656, −10.52084529364635403647991600238, −9.655353408254114219468410564534, −8.390103782988061281440672287091, −7.35542124982747217536942964879, −5.90272646061849900067791870693, −5.45264683168649488597880600402, −4.07095494634357691781789485927, −3.17630320324709603901775553482, −1.73531478260034522505707607493,
0.836398417349163977336429994510, 2.30904528340353133065959990325, 3.57023772329820843340289055237, 4.73790245462222044729579412975, 5.86472703774097061989539079065, 6.94779043284546955180712781256, 7.53131743319261212690278803697, 8.846189966142124708580789451114, 10.29436423034775032329372184739, 11.07956209384122394021994152111