| L(s) = 1 | + 10.0i·2-s + 50.8·3-s + 26.8·4-s + 8.88i·5-s + 511. i·6-s + 510. i·7-s + 1.55e3i·8-s + 400.·9-s − 89.3·10-s − 4.06e3i·11-s + 1.36e3·12-s + (2.77e3 − 7.42e3i)13-s − 5.13e3·14-s + 452. i·15-s − 1.22e4·16-s − 1.79e3·17-s + ⋯ |
| L(s) = 1 | + 0.889i·2-s + 1.08·3-s + 0.209·4-s + 0.0317i·5-s + 0.966i·6-s + 0.562i·7-s + 1.07i·8-s + 0.183·9-s − 0.0282·10-s − 0.920i·11-s + 0.227·12-s + (0.349 − 0.936i)13-s − 0.500·14-s + 0.0345i·15-s − 0.746·16-s − 0.0884·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.74753 + 1.21262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74753 + 1.21262i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.77e3 + 7.42e3i)T \) |
| good | 2 | \( 1 - 10.0iT - 128T^{2} \) |
| 3 | \( 1 - 50.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 8.88iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 510. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 4.06e3iT - 1.94e7T^{2} \) |
| 17 | \( 1 + 1.79e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.97e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.17e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.83e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.47e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 2.03e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 4.19e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 1.74e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.17e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.32e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.01e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 3.28e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.10e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 4.22e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 3.29e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 3.79e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.36e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 8.72e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 9.21e6iT - 8.07e13T^{2} \) |
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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
−18.54060879586410718637771561096, −16.88992250113775119381766027745, −15.51132680518883861520202279965, −14.70675983524644365495005446662, −13.35479683340856963807081646450, −11.22655847541079810010855974463, −8.907596455195162762939754818786, −7.81945150775814257357244166822, −5.85903372389327096984658518377, −2.80224878401327358655124585665,
1.92684499341758992252271506019, 3.73335292189720968711042580781, 7.17942133829713314136233129354, 9.129547907601130901687093152870, 10.59783297627341386921488988349, 12.21895223316106100875744540511, 13.68869080915865983674434398980, 14.99419547881236791714827663076, 16.58290631344701733456281612137, 18.54462620299598119344898649261
