| L(s) = 1 | + (−2.45 − 3.07i)2-s + (2.79 − 1.34i)3-s + (−1.66 + 7.30i)4-s + (16.4 − 7.93i)5-s + (−10.9 − 5.28i)6-s + (−6.70 − 17.2i)7-s + (−1.80 + 0.871i)8-s + (−10.8 + 13.6i)9-s + (−64.8 − 31.2i)10-s + (6.93 + 8.69i)11-s + (5.16 + 22.6i)12-s + (−20.0 − 25.1i)13-s + (−36.6 + 62.9i)14-s + (35.3 − 44.3i)15-s + (61.1 + 29.4i)16-s + (−15.1 − 66.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.867 − 1.08i)2-s + (0.536 − 0.258i)3-s + (−0.208 + 0.912i)4-s + (1.47 − 0.710i)5-s + (−0.747 − 0.359i)6-s + (−0.361 − 0.932i)7-s + (−0.0799 + 0.0385i)8-s + (−0.402 + 0.504i)9-s + (−2.05 − 0.988i)10-s + (0.190 + 0.238i)11-s + (0.124 + 0.544i)12-s + (−0.427 − 0.535i)13-s + (−0.700 + 1.20i)14-s + (0.608 − 0.762i)15-s + (0.954 + 0.459i)16-s + (−0.216 − 0.949i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.484878 - 1.04484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.484878 - 1.04484i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (6.70 + 17.2i)T \) |
| good | 2 | \( 1 + (2.45 + 3.07i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (-2.79 + 1.34i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (-16.4 + 7.93i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (-6.93 - 8.69i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (20.0 + 25.1i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (15.1 + 66.5i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (33.6 - 147. i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-40.8 - 179. i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (56.8 + 249. i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (-214. + 103. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-81.3 - 39.1i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-220. - 276. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (51.4 - 225. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (215. + 103. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (7.36 + 32.2i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 + 505.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (152. - 666. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-366. + 459. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + 50.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-348. + 437. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (730. - 915. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 - 5.56T + 9.12e5T^{2} \) |
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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
−14.12345158815450814022636864223, −13.53068839178751961296981677137, −12.35324067019877407184754809933, −10.83209328677430640486443517236, −9.713692145045595376976005224746, −9.169830634872143963602783904679, −7.57726506767287956927086250007, −5.45143531769039461404640501922, −2.79293207738768927686340020412, −1.25142546321271291338610560729,
2.74236424569100957374869731312, 5.90364134770564412261242256087, 6.55715097602727783806791837531, 8.398005623348579564060269010443, 9.375760499435190062002058863448, 10.02695885005472655444758062537, 12.06215102358505770058092861434, 13.79625300173272047873533694568, 14.66495973865871552003406249037, 15.46110021059769741177545000720
