| L(s) = 1 | + 2.98i·2-s − 41.6i·3-s + 55.0·4-s − 126. i·5-s + 124.·6-s − 69.5i·7-s + 355. i·8-s − 1.00e3·9-s + 377.·10-s − 100.·11-s − 2.29e3i·12-s − 1.72e3·13-s + 207.·14-s − 5.26e3·15-s + 2.46e3·16-s − 115.·17-s + ⋯ |
| L(s) = 1 | + 0.373i·2-s − 1.54i·3-s + 0.860·4-s − 1.01i·5-s + 0.576·6-s − 0.202i·7-s + 0.694i·8-s − 1.38·9-s + 0.377·10-s − 0.0755·11-s − 1.32i·12-s − 0.784·13-s + 0.0757·14-s − 1.55·15-s + 0.600·16-s − 0.0235·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.987282 - 1.58710i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987282 - 1.58710i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.51e4 + 7.13e4i)T \) |
| good | 2 | \( 1 - 2.98iT - 64T^{2} \) |
| 3 | \( 1 + 41.6iT - 729T^{2} \) |
| 5 | \( 1 + 126. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 69.5iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 100.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.72e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 115.T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.21e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 6.39e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.54e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.20e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 3.82e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 7.86e4T + 4.75e9T^{2} \) |
| 47 | \( 1 - 1.38e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 5.48e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.77e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.72e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 5.75e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.67e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.75e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.34e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 6.87e4T + 3.26e11T^{2} \) |
| 89 | \( 1 - 3.35e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.41e5T + 8.32e11T^{2} \) |
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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
−14.17549312120963406695569967160, −12.91885224863799334627260205957, −12.27357534493781424544963875211, −11.05633506613531655808528082570, −8.909887583219931842098610444857, −7.58886112838320693604978238026, −6.82369676547416128864386079240, −5.30327627775963521302825880210, −2.34822331850471858502580306668, −0.879621182773044438739575679714,
2.59789374962522858603354960214, 3.86293532430408012273696120175, 5.76232027245710540873234905194, 7.39564307588810387311602488731, 9.428738587826980301286594374337, 10.45507809413341435973268911678, 11.04395356077957473366834640145, 12.34383034786804134336795989213, 14.48966760215397273560433256303, 15.04808736370064424716256854015
