| L(s) = 1 | + (22.3 + 38.7i)2-s + (−79.2 − 137. i)3-s + (25.1 − 43.5i)4-s + 2.00e3·5-s + (3.54e3 − 6.13e3i)6-s + (1.24e4 − 2.15e4i)7-s + 9.37e4·8-s + (7.60e4 − 1.31e5i)9-s + (4.47e4 + 7.75e4i)10-s + (1.74e5 + 3.02e5i)11-s − 7.96e3·12-s + (5.93e5 + 1.19e6i)13-s + 1.10e6·14-s + (−1.58e5 − 2.74e5i)15-s + (2.04e6 + 3.54e6i)16-s + (1.98e6 − 3.43e6i)17-s + ⋯ |
| L(s) = 1 | + (0.493 + 0.855i)2-s + (−0.188 − 0.326i)3-s + (0.0122 − 0.0212i)4-s + 0.286·5-s + (0.185 − 0.322i)6-s + (0.279 − 0.483i)7-s + 1.01·8-s + (0.429 − 0.743i)9-s + (0.141 + 0.245i)10-s + (0.327 + 0.566i)11-s − 0.00924·12-s + (0.443 + 0.896i)13-s + 0.551·14-s + (−0.0539 − 0.0934i)15-s + (0.487 + 0.844i)16-s + (0.338 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.56611 + 0.262656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56611 + 0.262656i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-5.93e5 - 1.19e6i)T \) |
| good | 2 | \( 1 + (-22.3 - 38.7i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 3 | \( 1 + (79.2 + 137. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 - 2.00e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + (-1.24e4 + 2.15e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-1.74e5 - 3.02e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 17 | \( 1 + (-1.98e6 + 3.43e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-5.06e6 + 8.77e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (5.19e6 + 9.00e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (2.76e7 + 4.78e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 - 8.09e6T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-9.15e7 - 1.58e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-2.15e8 - 3.73e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (6.14e8 - 1.06e9i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 1.28e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.46e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (1.95e9 - 3.38e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (4.44e9 - 7.70e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-5.05e9 - 8.75e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + (-1.11e10 + 1.92e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 - 6.82e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.83e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.54e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.50e9 + 2.60e9i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (-7.72e10 + 1.33e11i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−17.01147656742875712929100385999, −15.70679282218071806214427997757, −14.43611154192166666450444730007, −13.34238376468796824845007397651, −11.55620715705912845246991118408, −9.693640117091080516626036670789, −7.36352320841354421047978873745, −6.26325135025077121469974114282, −4.43923855432155068821131285275, −1.35768454606050820025220099945,
1.75363175585045025372324478029, 3.62107549646566596050601162520, 5.46877892162314839961083875867, 7.982687482863965264319638163376, 10.17326947513392601238775425711, 11.33535093830585659732378589476, 12.70495317687172909131563180666, 13.94621457630747879410827536913, 15.77363084062332637638196396988, 17.02027498878412112381381838866
