| L(s) = 1 | + (−0.0729 − 1.39i)2-s + (2.18 + 1.77i)3-s + (0.0566 − 0.00595i)4-s + (−0.287 − 2.21i)5-s + (2.30 − 3.17i)6-s + (−2.62 + 0.311i)7-s + (−0.448 − 2.83i)8-s + (1.02 + 4.82i)9-s + (−3.06 + 0.562i)10-s + (1.26 + 0.268i)11-s + (0.134 + 0.0873i)12-s + (1.23 + 2.42i)13-s + (0.624 + 3.63i)14-s + (3.30 − 5.36i)15-s + (−3.79 + 0.807i)16-s + (1.66 + 0.639i)17-s + ⋯ |
| L(s) = 1 | + (−0.0515 − 0.984i)2-s + (1.26 + 1.02i)3-s + (0.0283 − 0.00297i)4-s + (−0.128 − 0.991i)5-s + (0.942 − 1.29i)6-s + (−0.993 + 0.117i)7-s + (−0.158 − 1.00i)8-s + (0.341 + 1.60i)9-s + (−0.969 + 0.177i)10-s + (0.380 + 0.0808i)11-s + (0.0388 + 0.0252i)12-s + (0.342 + 0.672i)13-s + (0.167 + 0.971i)14-s + (0.852 − 1.38i)15-s + (−0.949 + 0.201i)16-s + (0.403 + 0.155i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47935 - 0.621663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47935 - 0.621663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.287 + 2.21i)T \) |
| 7 | \( 1 + (2.62 - 0.311i)T \) |
| good | 2 | \( 1 + (0.0729 + 1.39i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-2.18 - 1.77i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.26 - 0.268i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 2.42i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.66 - 0.639i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.525 - 4.99i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (7.99 - 0.419i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.927 - 1.27i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.58 + 8.05i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (1.62 - 2.50i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-11.6 - 3.78i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (4.19 + 4.19i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.18 + 5.68i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (2.51 - 3.10i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (2.98 + 3.31i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-6.76 - 6.08i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 4.11i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-6.77 + 4.91i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.04 - 1.97i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (1.50 - 3.38i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.0807 + 0.0127i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-10.7 + 11.9i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-5.33 - 0.844i)T + (92.2 + 29.9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.50858218549619572768439201253, −11.67136484487784962927691246068, −10.22882409664185781641857225674, −9.670866678859924264192571862619, −8.993647666307694606127455277454, −7.87060022859823523963572984856, −6.07865377969781415250386143307, −4.10988781139594189266387563211, −3.59856069422125932089156893034, −2.03410171525777927263358635214,
2.45476617106814728836594947615, 3.44577640357779218293343461334, 6.03669360331535512403184403425, 6.80429568534588143541461286925, 7.50781966016923249028267675469, 8.358100944320597604258083286509, 9.472515163133284369208131138043, 10.81761015851968560428832156913, 12.12888035413503956771202331496, 13.14388624911349000937303895287
