| L(s) = 1 | + (1.99 − 0.648i)2-s + (−0.448 − 0.616i)3-s + (1.94 − 1.41i)4-s + (−0.794 − 2.09i)5-s + (−1.29 − 0.940i)6-s + i·7-s + (0.496 − 0.683i)8-s + (0.747 − 2.30i)9-s + (−2.94 − 3.65i)10-s + (1.57 + 4.83i)11-s + (−1.74 − 0.565i)12-s + (2.02 + 0.658i)13-s + (0.648 + 1.99i)14-s + (−0.932 + 1.42i)15-s + (−0.936 + 2.88i)16-s + (−1.80 + 2.47i)17-s + ⋯ |
| L(s) = 1 | + (1.41 − 0.458i)2-s + (−0.258 − 0.356i)3-s + (0.971 − 0.706i)4-s + (−0.355 − 0.934i)5-s + (−0.528 − 0.383i)6-s + 0.377i·7-s + (0.175 − 0.241i)8-s + (0.249 − 0.766i)9-s + (−0.930 − 1.15i)10-s + (0.474 + 1.45i)11-s + (−0.502 − 0.163i)12-s + (0.562 + 0.182i)13-s + (0.173 + 0.533i)14-s + (−0.240 + 0.368i)15-s + (−0.234 + 0.720i)16-s + (−0.436 + 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.72964 - 1.05481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72964 - 1.05481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.794 + 2.09i)T \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 + (-1.99 + 0.648i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.448 + 0.616i)T + (-0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (-1.57 - 4.83i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.02 - 0.658i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.80 - 2.47i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.68 + 1.22i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.532 + 0.173i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.34 + 4.61i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.40 + 3.20i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (8.55 + 2.77i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.29 + 3.97i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.14iT - 43T^{2} \) |
| 47 | \( 1 + (-5.56 - 7.65i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (5.66 + 7.79i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.71 + 8.36i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.61 - 8.03i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.99 + 6.87i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-8.39 + 6.10i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (11.3 - 3.68i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.91 - 4.29i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.59 - 6.32i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.850 - 2.61i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (8.25 + 11.3i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−12.53336362234468383763452013310, −12.05702437466877791256789172146, −11.12953846540096880399556153305, −9.561083127734177473297737693939, −8.526939083791547424293998603015, −6.91293877857719546822683197328, −5.88537710811839496301742134246, −4.62198749054292782637558482797, −3.86362988791825045889999703155, −1.86926723503030733772586010332,
3.11453555213572772527049729273, 4.05271410499498204363970913389, 5.30238845589866065237852552794, 6.37638397422144430109661819240, 7.24005969729193583542844570053, 8.620329753310963995031052856505, 10.37085813070330357403717918499, 11.06395147100424595015087193580, 11.94016359740498095230327957483, 13.25902319115149778863682437939
