| 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
−15.45304687228421403797547808245, −13.73492510516019618186199873557, −12.04598130114582165573826130692, −11.30543984607989649069629476302, −10.71724364473593766280138777941, −9.467861509105215100784206936714, −8.243279826298309786905823998529, −5.97325923018157398577866701360, −4.21526768698200161772708571733, −1.53661526104010895579617854174,
0.75859643704366725058430304390, 5.36290095275637465313767784799, 6.25582383844106841867861881842, 7.56311902325719679067352385749, 8.563426667030047778544604819576, 10.30315284462839915508857606511, 11.35413686650418550203058660302, 12.73154234644504371702694544214, 14.29041411813623287691601132922, 15.48658753437293677793333874223
