L(s) = 1 | + (1.5 + 2.59i)3-s + (3.36 − 5.83i)5-s + (−11.2 + 14.7i)7-s + (−4.5 + 7.79i)9-s + (26.8 + 46.4i)11-s + 1.73·13-s + 20.2·15-s + (23.4 + 40.6i)17-s + (10.0 − 17.4i)19-s + (−55.1 − 7.10i)21-s + (−59.4 + 103. i)23-s + (39.8 + 68.9i)25-s − 27·27-s − 103.·29-s + (78.7 + 136. i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.301 − 0.521i)5-s + (−0.606 + 0.794i)7-s + (−0.166 + 0.288i)9-s + (0.735 + 1.27i)11-s + 0.0370·13-s + 0.347·15-s + (0.334 + 0.580i)17-s + (0.121 − 0.210i)19-s + (−0.572 − 0.0738i)21-s + (−0.539 + 0.933i)23-s + (0.318 + 0.551i)25-s − 0.192·27-s − 0.663·29-s + (0.456 + 0.790i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0648 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0648 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.24434 + 1.16607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24434 + 1.16607i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (11.2 - 14.7i)T \) |
good | 5 | \( 1 + (-3.36 + 5.83i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-26.8 - 46.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 1.73T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-23.4 - 40.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-10.0 + 17.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (59.4 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-78.7 - 136. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-18.8 + 32.6i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-110. + 190. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (146. + 253. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-297. - 516. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-132. + 229. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (468. + 811. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 545.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (149. + 259. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-470. + 814. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 611.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-588. + 1.01e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−12.52710517653473720126366305216, −11.77374112606812147035844927900, −10.29497065193291410506644897672, −9.425597204679149622267515815888, −8.840384383041143502844814117044, −7.39263927165586063750492912785, −6.02731707305570619000326008367, −4.89247121082086081964756986731, −3.53229044572082582385760570616, −1.86037056881270356515241278859,
0.791082907314467231340483911723, 2.76588164870414093192336457608, 3.96052603486089578250394012369, 5.96447954199102087643182302365, 6.72908117699135860307815230480, 7.85305991945285947568860687789, 9.033979743666679156369892274538, 10.10189182606494935897088993841, 11.07433462919022285521698025737, 12.13682268978806284258307342076