| L(s) = 1 | + (−2.77 + 3.47i)2-s + (−6.15 − 2.96i)3-s + (−2.61 − 11.4i)4-s + (1.98 + 0.955i)5-s + (27.3 − 13.1i)6-s + (18.5 + 0.0393i)7-s + (15.0 + 7.22i)8-s + (12.3 + 15.4i)9-s + (−8.81 + 4.24i)10-s + (−2.32 + 2.91i)11-s + (−17.8 + 78.3i)12-s + (55.9 − 70.2i)13-s + (−51.4 + 64.2i)14-s + (−9.38 − 11.7i)15-s + (17.9 − 8.66i)16-s + (12.4 − 54.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.979 + 1.22i)2-s + (−1.18 − 0.570i)3-s + (−0.326 − 1.43i)4-s + (0.177 + 0.0854i)5-s + (1.86 − 0.896i)6-s + (0.999 + 0.00212i)7-s + (0.663 + 0.319i)8-s + (0.455 + 0.571i)9-s + (−0.278 + 0.134i)10-s + (−0.0637 + 0.0799i)11-s + (−0.429 + 1.88i)12-s + (1.19 − 1.49i)13-s + (−0.982 + 1.22i)14-s + (−0.161 − 0.202i)15-s + (0.281 − 0.135i)16-s + (0.177 − 0.777i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.556009 - 0.0368840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.556009 - 0.0368840i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-18.5 - 0.0393i)T \) |
| good | 2 | \( 1 + (2.77 - 3.47i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (6.15 + 2.96i)T + (16.8 + 21.1i)T^{2} \) |
| 5 | \( 1 + (-1.98 - 0.955i)T + (77.9 + 97.7i)T^{2} \) |
| 11 | \( 1 + (2.32 - 2.91i)T + (-296. - 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-55.9 + 70.2i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-12.4 + 54.5i)T + (-4.42e3 - 2.13e3i)T^{2} \) |
| 19 | \( 1 + 27.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (12.4 + 54.3i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 29 | \( 1 + (22.1 - 97.0i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 - 99.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-79.7 + 349. i)T + (-4.56e4 - 2.19e4i)T^{2} \) |
| 41 | \( 1 + (-300. - 144. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (383. - 184. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-301. + 377. i)T + (-2.31e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (90.7 + 397. i)T + (-1.34e5 + 6.45e4i)T^{2} \) |
| 59 | \( 1 + (515. - 248. i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-34.2 + 150. i)T + (-2.04e5 - 9.84e4i)T^{2} \) |
| 67 | \( 1 - 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-86.0 - 377. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-422. - 530. i)T + (-8.65e4 + 3.79e5i)T^{2} \) |
| 79 | \( 1 + 677.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (494. + 619. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (80.1 + 100. i)T + (-1.56e5 + 6.87e5i)T^{2} \) |
| 97 | \( 1 - 430.T + 9.12e5T^{2} \) |
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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
−15.48658753437293677793333874223, −14.29041411813623287691601132922, −12.73154234644504371702694544214, −11.35413686650418550203058660302, −10.30315284462839915508857606511, −8.563426667030047778544604819576, −7.56311902325719679067352385749, −6.25582383844106841867861881842, −5.36290095275637465313767784799, −0.75859643704366725058430304390,
1.53661526104010895579617854174, 4.21526768698200161772708571733, 5.97325923018157398577866701360, 8.243279826298309786905823998529, 9.467861509105215100784206936714, 10.71724364473593766280138777941, 11.30543984607989649069629476302, 12.04598130114582165573826130692, 13.73492510516019618186199873557, 15.45304687228421403797547808245
