| 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
−15.46110021059769741177545000720, −14.66495973865871552003406249037, −13.79625300173272047873533694568, −12.06215102358505770058092861434, −10.02695885005472655444758062537, −9.375760499435190062002058863448, −8.398005623348579564060269010443, −6.55715097602727783806791837531, −5.90364134770564412261242256087, −2.74236424569100957374869731312,
1.25142546321271291338610560729, 2.79293207738768927686340020412, 5.45143531769039461404640501922, 7.57726506767287956927086250007, 9.169830634872143963602783904679, 9.713692145045595376976005224746, 10.83209328677430640486443517236, 12.35324067019877407184754809933, 13.53068839178751961296981677137, 14.12345158815450814022636864223
