| L(s) = 1 | + (12.5 − 3.35i)2-s + (−1.31 + 2.27i)3-s + (89.8 − 51.8i)4-s + (24.9 + 24.9i)5-s + (−8.79 + 32.8i)6-s + (−400. − 107. i)7-s + (363. − 363. i)8-s + (361. + 625. i)9-s + (395. + 228. i)10-s + (−169. − 631. i)11-s + 272. i·12-s + (−1.80e3 + 1.24e3i)13-s − 5.36e3·14-s + (−89.4 + 23.9i)15-s + (10.7 − 18.6i)16-s + (4.04e3 − 2.33e3i)17-s + ⋯ |
| L(s) = 1 | + (1.56 − 0.418i)2-s + (−0.0486 + 0.0841i)3-s + (1.40 − 0.810i)4-s + (0.199 + 0.199i)5-s + (−0.0407 + 0.151i)6-s + (−1.16 − 0.312i)7-s + (0.710 − 0.710i)8-s + (0.495 + 0.857i)9-s + (0.395 + 0.228i)10-s + (−0.127 − 0.474i)11-s + 0.157i·12-s + (−0.822 + 0.568i)13-s − 1.95·14-s + (−0.0264 + 0.00710i)15-s + (0.00262 − 0.00454i)16-s + (0.823 − 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.63579 - 0.557528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63579 - 0.557528i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.80e3 - 1.24e3i)T \) |
| good | 2 | \( 1 + (-12.5 + 3.35i)T + (55.4 - 32i)T^{2} \) |
| 3 | \( 1 + (1.31 - 2.27i)T + (-364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (-24.9 - 24.9i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + (400. + 107. i)T + (1.01e5 + 5.88e4i)T^{2} \) |
| 11 | \( 1 + (169. + 631. i)T + (-1.53e6 + 8.85e5i)T^{2} \) |
| 17 | \( 1 + (-4.04e3 + 2.33e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-2.58e3 + 9.66e3i)T + (-4.07e7 - 2.35e7i)T^{2} \) |
| 23 | \( 1 + (-1.38e4 - 7.98e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-6.61e3 + 1.14e4i)T + (-2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (6.05e3 + 6.05e3i)T + 8.87e8iT^{2} \) |
| 37 | \( 1 + (-2.10e4 - 7.84e4i)T + (-2.22e9 + 1.28e9i)T^{2} \) |
| 41 | \( 1 + (1.14e5 - 3.06e4i)T + (4.11e9 - 2.37e9i)T^{2} \) |
| 43 | \( 1 + (3.86e4 - 2.22e4i)T + (3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-3.40e4 + 3.40e4i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 + 1.46e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + (1.01e5 + 2.73e4i)T + (3.65e10 + 2.10e10i)T^{2} \) |
| 61 | \( 1 + (-1.07e5 - 1.85e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-4.70e5 + 1.26e5i)T + (7.83e10 - 4.52e10i)T^{2} \) |
| 71 | \( 1 + (-4.93e3 + 1.84e4i)T + (-1.10e11 - 6.40e10i)T^{2} \) |
| 73 | \( 1 + (-1.58e5 + 1.58e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 + 2.80e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (6.62e5 + 6.62e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + (4.80e3 + 1.79e4i)T + (-4.30e11 + 2.48e11i)T^{2} \) |
| 97 | \( 1 + (2.95e5 - 1.10e6i)T + (-7.21e11 - 4.16e11i)T^{2} \) |
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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
−18.97989700792454693046813824439, −16.67612795635632479986720840853, −15.42813979117905135824408865485, −13.82878042101889397414160075888, −13.10283314819927842766105928386, −11.57123327852619920052434813203, −9.974427987045679578031678802172, −6.80820595220534986423685446321, −4.93376219448312407065543485732, −2.94658798675343089936114800787,
3.43159906759084694609680032827, 5.52615242125628885299098096403, 6.98637086437242340352945412564, 9.775081336179492267748737288034, 12.44842132678913635485404775946, 12.72786104119378318083123562195, 14.52867145562776349338401447099, 15.51810360484385998754452753431, 16.81469291003899860284901308174, 18.72688913869080207417316232146
