| 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 | \( 1 + (-5.33 + 0.844i)T + (92.2 - 29.9i)T^{2} \) |
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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
−13.14388624911349000937303895287, −12.12888035413503956771202331496, −10.81761015851968560428832156913, −9.472515163133284369208131138043, −8.358100944320597604258083286509, −7.50781966016923249028267675469, −6.80429568534588143541461286925, −6.03669360331535512403184403425, −3.44577640357779218293343461334, −2.45476617106814728836594947615,
2.03410171525777927263358635214, 3.59856069422125932089156893034, 4.10988781139594189266387563211, 6.07865377969781415250386143307, 7.87060022859823523963572984856, 8.993647666307694606127455277454, 9.670866678859924264192571862619, 10.22882409664185781641857225674, 11.67136484487784962927691246068, 12.50858218549619572768439201253
