| L(s) = 1 | + (−1.80 + 4.58i)2-s + (34.7 + 2.60i)3-s + (29.1 + 27.0i)4-s + (−64.1 + 94.1i)5-s + (−74.5 + 154. i)6-s + (−320. + 121. i)7-s + (−460. + 221. i)8-s + (480. + 72.4i)9-s + (−316. − 464. i)10-s + (1.28e3 − 193. i)11-s + (941. + 1.01e3i)12-s + (−604. + 481. i)13-s + (20.7 − 1.69e3i)14-s + (−2.47e3 + 3.10e3i)15-s + (1.70 + 22.7i)16-s + (347. − 1.12e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.573i)2-s + (1.28 + 0.0964i)3-s + (0.454 + 0.422i)4-s + (−0.513 + 0.753i)5-s + (−0.345 + 0.716i)6-s + (−0.935 + 0.353i)7-s + (−0.899 + 0.433i)8-s + (0.658 + 0.0993i)9-s + (−0.316 − 0.464i)10-s + (0.965 − 0.145i)11-s + (0.544 + 0.587i)12-s + (−0.275 + 0.219i)13-s + (0.00756 − 0.615i)14-s + (−0.733 + 0.920i)15-s + (0.000416 + 0.00555i)16-s + (0.0708 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.711265 + 1.89213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.711265 + 1.89213i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (320. - 121. i)T \) |
| good | 2 | \( 1 + (1.80 - 4.58i)T + (-46.9 - 43.5i)T^{2} \) |
| 3 | \( 1 + (-34.7 - 2.60i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (64.1 - 94.1i)T + (-5.70e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-1.28e3 + 193. i)T + (1.69e6 - 5.22e5i)T^{2} \) |
| 13 | \( 1 + (604. - 481. i)T + (1.07e6 - 4.70e6i)T^{2} \) |
| 17 | \( 1 + (-347. + 1.12e3i)T + (-1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (-1.02e3 + 590. i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (1.74e3 - 538. i)T + (1.22e8 - 8.33e7i)T^{2} \) |
| 29 | \( 1 + (-5.61e3 - 2.46e4i)T + (-5.35e8 + 2.58e8i)T^{2} \) |
| 31 | \( 1 + (-4.06e4 - 2.34e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-5.34e4 + 4.95e4i)T + (1.91e8 - 2.55e9i)T^{2} \) |
| 41 | \( 1 + (-1.77e4 - 3.68e4i)T + (-2.96e9 + 3.71e9i)T^{2} \) |
| 43 | \( 1 + (-4.18e4 - 2.01e4i)T + (3.94e9 + 4.94e9i)T^{2} \) |
| 47 | \( 1 + (272. + 106. i)T + (7.90e9 + 7.33e9i)T^{2} \) |
| 53 | \( 1 + (1.33e5 + 1.23e5i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 - 1.50e5i)T + (-1.54e10 + 3.92e10i)T^{2} \) |
| 61 | \( 1 + (-8.69e4 - 9.36e4i)T + (-3.85e9 + 5.13e10i)T^{2} \) |
| 67 | \( 1 + (-1.43e5 + 2.49e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-6.80e4 + 2.98e5i)T + (-1.15e11 - 5.55e10i)T^{2} \) |
| 73 | \( 1 + (-1.89e5 + 7.43e4i)T + (1.10e11 - 1.02e11i)T^{2} \) |
| 79 | \( 1 + (-4.37e5 - 7.58e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (5.61e4 + 4.47e4i)T + (7.27e10 + 3.18e11i)T^{2} \) |
| 89 | \( 1 + (-8.65e4 + 5.74e5i)T + (-4.74e11 - 1.46e11i)T^{2} \) |
| 97 | \( 1 + 1.66e6iT - 8.32e11T^{2} \) |
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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
−14.88706588925157291577446398530, −14.08965913255276586177254527052, −12.52711270888518416753103931603, −11.36067507512618871964995639331, −9.553073170284183848515668081370, −8.605180024281516564499815951608, −7.38995844244798337813773155103, −6.41771798729878372882120011999, −3.55254128062608715930835782591, −2.69471886959924232556500418305,
0.838447873344998246895174532288, 2.59165651896593133625192816555, 3.95942970389157203981171607287, 6.39614375414299586728053791433, 7.955013839937783192320118868995, 9.223371118033536312544698777940, 9.990238255154605121751474245044, 11.67783658763955655165564203561, 12.68225253207132397823370501911, 13.90418887477019101373833416810
