| L(s) = 1 | + (−13.5 + 23.5i)2-s + (40.2 − 69.6i)3-s + (−113. − 196. i)4-s + 1.95e3·5-s + (1.09e3 + 1.89e3i)6-s + (540. + 935. i)7-s − 7.76e3·8-s + (6.60e3 + 1.14e4i)9-s + (−2.65e4 + 4.59e4i)10-s + (−2.67e4 + 4.62e4i)11-s − 1.82e4·12-s + (1.00e5 + 2.18e4i)13-s − 2.93e4·14-s + (7.84e4 − 1.35e5i)15-s + (1.63e5 − 2.83e5i)16-s + (1.46e5 + 2.53e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.600 + 1.03i)2-s + (0.286 − 0.496i)3-s + (−0.221 − 0.382i)4-s + 1.39·5-s + (0.344 + 0.596i)6-s + (0.0850 + 0.147i)7-s − 0.670·8-s + (0.335 + 0.581i)9-s + (−0.838 + 1.45i)10-s + (−0.550 + 0.952i)11-s − 0.253·12-s + (0.977 + 0.211i)13-s − 0.204·14-s + (0.400 − 0.693i)15-s + (0.623 − 1.07i)16-s + (0.424 + 0.735i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0420 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0420 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.10998 + 1.15762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10998 + 1.15762i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.00e5 - 2.18e4i)T \) |
| good | 2 | \( 1 + (13.5 - 23.5i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-40.2 + 69.6i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 - 1.95e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + (-540. - 935. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.67e4 - 4.62e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-1.46e5 - 2.53e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (5.53e4 + 9.58e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-4.19e4 + 7.26e4i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-1.63e6 + 2.83e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 7.28e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (3.43e6 - 5.95e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-1.49e7 + 2.59e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.68e7 + 2.92e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 - 2.58e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.80e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-6.34e7 - 1.09e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (6.49e7 + 1.12e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.84e7 + 1.01e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-6.03e7 - 1.04e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 3.81e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.67e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.24e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (3.41e6 - 5.90e6i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-3.88e7 - 6.72e7i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.90214639930781517571133930706, −16.88707987667814581603017653865, −15.47919484832605859732085569955, −13.97758745546848368302730541161, −12.74712234375135913228622925413, −10.19449366076721203693347231942, −8.681760353449425335112778029388, −7.19361145066612580181767619652, −5.72422550415215691862815139496, −1.98007760705747429555050589623,
1.19917105593900284227260102818, 3.07844278333424991040018815907, 5.91752464776868292684212572944, 8.911236827749199925566562412517, 9.932386630447705053397889154568, 11.01430716475031939241494278002, 12.94538262198449334848266069933, 14.38039404569002637437680571880, 16.12233003031348340057121086991, 17.89201703419342325699020324548
