| L(s) = 1 | + (−25.8 + 18.8i)2-s + (−123. + 380. i)3-s + (316. − 973. i)4-s + (−4.41e3 + 5.41e3i)5-s + (−3.95e3 − 1.21e4i)6-s + 6.67e3·7-s + (1.01e4 + 3.11e4i)8-s + (1.40e4 + 1.02e4i)9-s + (1.25e4 − 2.23e5i)10-s + (−6.25e5 + 4.54e5i)11-s + (3.31e5 + 2.40e5i)12-s + (−8.15e5 − 5.92e5i)13-s + (−1.72e5 + 1.25e5i)14-s + (−1.51e6 − 2.34e6i)15-s + (−8.48e5 − 6.16e5i)16-s + (1.47e6 + 4.53e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.293 + 0.903i)3-s + (0.154 − 0.475i)4-s + (−0.632 + 0.774i)5-s + (−0.207 − 0.638i)6-s + 0.150·7-s + (0.109 + 0.336i)8-s + (0.0795 + 0.0577i)9-s + (0.0396 − 0.705i)10-s + (−1.17 + 0.850i)11-s + (0.384 + 0.279i)12-s + (−0.608 − 0.442i)13-s + (−0.0858 + 0.0623i)14-s + (−0.514 − 0.798i)15-s + (−0.202 − 0.146i)16-s + (0.251 + 0.775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0775190 - 0.0246075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0775190 - 0.0246075i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 - 18.8i)T \) |
| 5 | \( 1 + (4.41e3 - 5.41e3i)T \) |
| good | 3 | \( 1 + (123. - 380. i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 7 | \( 1 - 6.67e3T + 1.97e9T^{2} \) |
| 11 | \( 1 + (6.25e5 - 4.54e5i)T + (8.81e10 - 2.71e11i)T^{2} \) |
| 13 | \( 1 + (8.15e5 + 5.92e5i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-1.47e6 - 4.53e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (5.53e6 + 1.70e7i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 + (4.46e7 - 3.24e7i)T + (2.94e14 - 9.06e14i)T^{2} \) |
| 29 | \( 1 + (2.55e7 - 7.85e7i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (3.16e7 + 9.75e7i)T + (-2.05e16 + 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-5.58e8 - 4.05e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-1.07e8 - 7.84e7i)T + (1.70e17 + 5.23e17i)T^{2} \) |
| 43 | \( 1 - 1.36e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-8.98e8 + 2.76e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-5.59e8 + 1.72e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-7.37e9 - 5.36e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-4.29e9 + 3.11e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 + (-2.48e9 - 7.63e9i)T + (-9.88e19 + 7.17e19i)T^{2} \) |
| 71 | \( 1 + (5.26e9 - 1.62e10i)T + (-1.86e20 - 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-1.42e10 + 1.03e10i)T + (9.69e19 - 2.98e20i)T^{2} \) |
| 79 | \( 1 + (3.78e9 - 1.16e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (1.90e10 + 5.87e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + (5.27e10 - 3.83e10i)T + (8.57e20 - 2.63e21i)T^{2} \) |
| 97 | \( 1 + (1.19e10 - 3.66e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−13.05697588389617782961722312008, −11.43837406484104763330309298916, −10.45524932742183013226637813106, −9.790479260508266859198411960794, −8.027211244125671237152297288414, −7.14096817344449511616614272437, −5.41973723209251421780105980193, −4.18318385527641586568520341754, −2.37635633768520427035955923733, −0.03909061775070004707562079576,
0.856087513000627979892632249298, 2.27125503524713754250833478466, 4.14775277458402344898821698598, 5.89690161557656602575309774491, 7.58589716463040553296429057648, 8.190563172340737995657644511245, 9.712607585826414096309976741026, 11.12024774910350972651726766072, 12.27247597310925764609404560666, 12.72718336492727592809366357073
