L(s) = 1 | + (29.4 + 56.8i)2-s + 420. i·3-s + (−2.35e3 + 3.34e3i)4-s + 2.89e4·5-s + (−2.39e4 + 1.24e4i)6-s + 1.57e5i·7-s + (−2.59e5 − 3.51e4i)8-s − 1.77e5·9-s + (8.53e5 + 1.64e6i)10-s − 1.55e6i·11-s + (−1.40e6 − 9.92e5i)12-s − 4.64e6·13-s + (−8.97e6 + 4.65e6i)14-s + 1.21e7i·15-s + (−5.65e6 − 1.57e7i)16-s + 9.00e6·17-s + ⋯ |
L(s) = 1 | + (0.460 + 0.887i)2-s + 0.577i·3-s + (−0.575 + 0.817i)4-s + 1.85·5-s + (−0.512 + 0.265i)6-s + 1.34i·7-s + (−0.990 − 0.134i)8-s − 0.333·9-s + (0.853 + 1.64i)10-s − 0.879i·11-s + (−0.472 − 0.332i)12-s − 0.962·13-s + (−1.19 + 0.618i)14-s + 1.06i·15-s + (−0.337 − 0.941i)16-s + 0.372·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.766054 + 2.41910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.766054 + 2.41910i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-29.4 - 56.8i)T \) |
| 3 | \( 1 - 420. iT \) |
good | 5 | \( 1 - 2.89e4T + 2.44e8T^{2} \) |
| 7 | \( 1 - 1.57e5iT - 1.38e10T^{2} \) |
| 11 | \( 1 + 1.55e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 + 4.64e6T + 2.32e13T^{2} \) |
| 17 | \( 1 - 9.00e6T + 5.82e14T^{2} \) |
| 19 | \( 1 - 4.57e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 7.97e6iT - 2.19e16T^{2} \) |
| 29 | \( 1 - 8.09e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + 4.31e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 1.23e9T + 6.58e18T^{2} \) |
| 41 | \( 1 - 3.60e9T + 2.25e19T^{2} \) |
| 43 | \( 1 - 1.36e8iT - 3.99e19T^{2} \) |
| 47 | \( 1 + 5.87e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 - 6.42e9T + 4.91e20T^{2} \) |
| 59 | \( 1 - 4.44e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 1.83e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 6.73e10iT - 8.18e21T^{2} \) |
| 71 | \( 1 - 9.67e10iT - 1.64e22T^{2} \) |
| 73 | \( 1 - 1.46e11T + 2.29e22T^{2} \) |
| 79 | \( 1 - 9.30e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + 5.31e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 1.05e11T + 2.46e23T^{2} \) |
| 97 | \( 1 - 7.32e11T + 6.93e23T^{2} \) |
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
−17.41900621874252318932976245780, −16.34108874622826152937617978756, −14.81020819556826195469778693317, −13.85895898844825468970223539979, −12.33381070290364421653461683536, −9.859036885942647308383590660547, −8.704374635772965845468197892154, −6.08912504123053581651051084479, −5.28594498703015877355729623494, −2.65156154114481790958712462019,
1.08220836331437998488158736278, 2.43680522867133081611075224131, 4.95627984622138324219539254014, 6.74894212580766195607638580757, 9.571375440584077733694297480447, 10.51564475775248391710284525546, 12.56366369697338759836153907927, 13.62111477662230257702905921204, 14.38528662836449245808053488372, 17.28848113481449778721436439638