| L(s) = 1 | + (−2.99 − 11.1i)2-s + (−49.8 + 86.3i)3-s + (105. − 61.1i)4-s + (403. − 403. i)5-s + (1.11e3 + 298. i)6-s + (636. − 2.37e3i)7-s + (−3.09e3 − 3.09e3i)8-s + (−1.69e3 − 2.93e3i)9-s + (−5.71e3 − 3.30e3i)10-s + (1.82e4 − 4.88e3i)11-s + 1.22e4i·12-s + (2.59e4 − 1.18e4i)13-s − 2.84e4·14-s + (1.47e4 + 5.50e4i)15-s + (−9.62e3 + 1.66e4i)16-s + (−6.93e4 + 4.00e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.187 − 0.697i)2-s + (−0.615 + 1.06i)3-s + (0.413 − 0.238i)4-s + (0.646 − 0.646i)5-s + (0.859 + 0.230i)6-s + (0.264 − 0.988i)7-s + (−0.755 − 0.755i)8-s + (−0.258 − 0.447i)9-s + (−0.571 − 0.330i)10-s + (1.24 − 0.333i)11-s + 0.588i·12-s + (0.909 − 0.414i)13-s − 0.739·14-s + (0.291 + 1.08i)15-s + (−0.146 + 0.254i)16-s + (−0.830 + 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.22555 - 0.821947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22555 - 0.821947i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.59e4 + 1.18e4i)T \) |
| good | 2 | \( 1 + (2.99 + 11.1i)T + (-221. + 128i)T^{2} \) |
| 3 | \( 1 + (49.8 - 86.3i)T + (-3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-403. + 403. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + (-636. + 2.37e3i)T + (-4.99e6 - 2.88e6i)T^{2} \) |
| 11 | \( 1 + (-1.82e4 + 4.88e3i)T + (1.85e8 - 1.07e8i)T^{2} \) |
| 17 | \( 1 + (6.93e4 - 4.00e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.12e5 + 3.00e4i)T + (1.47e10 + 8.49e9i)T^{2} \) |
| 23 | \( 1 + (-1.14e5 - 6.63e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (2.09e5 - 3.62e5i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.21e6 + 1.21e6i)T - 8.52e11iT^{2} \) |
| 37 | \( 1 + (4.04e5 - 1.08e5i)T + (3.04e12 - 1.75e12i)T^{2} \) |
| 41 | \( 1 + (-9.87e5 - 3.68e6i)T + (-6.91e12 + 3.99e12i)T^{2} \) |
| 43 | \( 1 + (-1.34e6 + 7.77e5i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-1.65e6 - 1.65e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + 7.38e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (3.33e6 - 1.24e7i)T + (-1.27e14 - 7.34e13i)T^{2} \) |
| 61 | \( 1 + (-1.15e7 - 2.00e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (3.42e6 + 1.27e7i)T + (-3.51e14 + 2.03e14i)T^{2} \) |
| 71 | \( 1 + (3.30e7 + 8.84e6i)T + (5.59e14 + 3.22e14i)T^{2} \) |
| 73 | \( 1 + (-1.87e6 - 1.87e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 6.81e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.10e7 - 2.10e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + (6.40e7 - 1.71e7i)T + (3.40e15 - 1.96e15i)T^{2} \) |
| 97 | \( 1 + (3.49e7 + 9.37e6i)T + (6.78e15 + 3.91e15i)T^{2} \) |
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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
−17.40122210371193692310653425355, −16.57681276216796376691896528473, −15.19257783719573941269263646195, −13.29034876653560983172114222793, −11.34500310109361316828092544518, −10.53080984471901586920671979096, −9.217618923135989585517410197953, −6.16014322934654751678092898157, −4.14718097732343406018714122940, −1.17008806436447655737043314718,
2.03593970038056618594119355161, 6.17102745894691541370272966607, 6.78136948887629325495722215745, 8.768727642968349221633755636660, 11.32394195483301369749488660665, 12.35193689679763062451508835975, 14.18661058265006914784112465556, 15.57175689338987274940897680743, 17.25132230658532805800421903718, 17.91802467524572409476741638169
