L(s) = 1 | + 20.0i·2-s + (−46.5 + 4.94i)3-s − 272.·4-s + 368.·5-s + (−98.8 − 930. i)6-s − 90.4i·7-s − 2.89e3i·8-s + (2.13e3 − 459. i)9-s + 7.36e3i·10-s + 7.61e3·11-s + (1.26e4 − 1.34e3i)12-s + 684.·13-s + 1.81e3·14-s + (−1.71e4 + 1.81e3i)15-s + 2.29e4·16-s + 1.71e4·17-s + ⋯ |
L(s) = 1 | + 1.76i·2-s + (−0.994 + 0.105i)3-s − 2.12·4-s + 1.31·5-s + (−0.186 − 1.75i)6-s − 0.0996i·7-s − 1.99i·8-s + (0.977 − 0.210i)9-s + 2.32i·10-s + 1.72·11-s + (2.11 − 0.224i)12-s + 0.0863·13-s + 0.176·14-s + (−1.30 + 0.139i)15-s + 1.40·16-s + 0.847·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.782074 + 1.56611i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782074 + 1.56611i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (46.5 - 4.94i)T \) |
| 23 | \( 1 + (-2.99e4 - 5.00e4i)T \) |
good | 2 | \( 1 - 20.0iT - 128T^{2} \) |
| 5 | \( 1 - 368.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 90.4iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 7.61e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 684.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.71e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.09e4iT - 8.93e8T^{2} \) |
| 29 | \( 1 + 1.61e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 4.07e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.72e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 4.12e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 4.34e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 9.50e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 3.96e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.08e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.86e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 1.08e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.31e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 1.75e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.62e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 5.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.03e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.54e7iT - 8.07e13T^{2} \) |
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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
−13.92225382266844877131105769928, −13.03208120084160394057645083422, −11.47674312886235521308387047780, −9.760314743403857590778390884869, −9.137538609217356398139216700909, −7.27925787368451091610794215004, −6.31423747270512110053262539089, −5.65457247787373401771600064591, −4.38399950338564255396602014167, −1.04224243127171755427465460999,
1.04116280130186001194415438733, 1.83253764112977725412718781615, 3.73038553268170278612410542630, 5.23865055628938161603152277829, 6.48602834757680694946128625894, 8.963102555766609577859671237600, 9.957121648866400199882037670858, 10.61328474281720088596580623404, 11.92725265323715941389113249697, 12.39515686118003229894309783415