| L(s) = 1 | + (−4.50 − 7.81i)3-s + (5.65 + 48.6i)7-s + (−0.179 + 0.311i)9-s + (72.8 + 126. i)11-s + 209.·13-s + (93.5 + 162. i)17-s + (−153. − 88.6i)19-s + (354. − 263. i)21-s + (−288. − 166. i)23-s − 727.·27-s + 1.38e3·29-s + (−1.34e3 + 776. i)31-s + (657. − 1.13e3i)33-s + (−1.97e3 − 1.13e3i)37-s + (−946. − 1.63e3i)39-s + ⋯ |
| L(s) = 1 | + (−0.501 − 0.867i)3-s + (0.115 + 0.993i)7-s + (−0.00222 + 0.00384i)9-s + (0.602 + 1.04i)11-s + 1.24·13-s + (0.323 + 0.560i)17-s + (−0.425 − 0.245i)19-s + (0.804 − 0.598i)21-s + (−0.546 − 0.315i)23-s − 0.997·27-s + 1.64·29-s + (−1.40 + 0.808i)31-s + (0.603 − 1.04i)33-s + (−1.44 − 0.831i)37-s + (−0.622 − 1.07i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.234020212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234020212\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-5.65 - 48.6i)T \) |
| good | 3 | \( 1 + (4.50 + 7.81i)T + (-40.5 + 70.1i)T^{2} \) |
| 11 | \( 1 + (-72.8 - 126. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 209.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-93.5 - 162. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (153. + 88.6i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (288. + 166. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 - 1.38e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (1.34e3 - 776. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.97e3 + 1.13e3i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 - 781. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 837. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (748. - 1.29e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (3.86e3 - 2.23e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.81e3 + 2.20e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.25e3 - 1.30e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (5.00e3 - 2.88e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 1.14e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (1.84e3 + 3.19e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.34e3 + 2.33e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 2.68e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (5.03e3 + 2.90e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 7.89e3T + 8.85e7T^{2} \) |
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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
−10.10200106694206550139826647348, −9.042747116551124385140286596027, −8.403717537616932805457122923739, −7.29021158710482696326586654230, −6.42766535422435829932853602677, −5.91006501491857597930851683053, −4.69876977254884902769531021900, −3.50152752871293747137839134766, −1.99725635461651181243845939515, −1.25365166954925303448680959248,
0.33168843631618051188287833601, 1.52509410466711084028451989781, 3.46005352089338137324079979942, 3.99357067200551983541106942316, 5.06824661888265399498281820188, 5.99497405046565798746301666506, 6.89383722989367026059286002470, 8.049742448410723172310598619425, 8.823196446979848072777309958568, 9.920707633848140129150734570677
