| L(s) = 1 | + (−0.683 − 1.78i)2-s + (0.100 − 1.92i)3-s + (−1.21 + 1.09i)4-s + (2.21 + 0.310i)5-s + (−3.49 + 1.13i)6-s + (−0.357 − 2.62i)7-s + (−0.615 − 0.313i)8-s + (−0.705 − 0.0741i)9-s + (−0.960 − 4.15i)10-s + (0.0828 + 0.788i)11-s + (1.98 + 2.45i)12-s + (0.692 + 4.36i)13-s + (−4.42 + 2.42i)14-s + (0.820 − 4.22i)15-s + (−0.480 + 4.56i)16-s + (1.83 − 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.483 − 1.25i)2-s + (0.0581 − 1.11i)3-s + (−0.608 + 0.547i)4-s + (0.990 + 0.138i)5-s + (−1.42 + 0.463i)6-s + (−0.135 − 0.990i)7-s + (−0.217 − 0.110i)8-s + (−0.235 − 0.0247i)9-s + (−0.303 − 1.31i)10-s + (0.0249 + 0.237i)11-s + (0.572 + 0.707i)12-s + (0.191 + 1.21i)13-s + (−1.18 + 0.648i)14-s + (0.211 − 1.09i)15-s + (−0.120 + 1.14i)16-s + (0.444 − 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.166067 - 1.03335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166067 - 1.03335i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.21 - 0.310i)T \) |
| 7 | \( 1 + (0.357 + 2.62i)T \) |
| good | 2 | \( 1 + (0.683 + 1.78i)T + (-1.48 + 1.33i)T^{2} \) |
| 3 | \( 1 + (-0.100 + 1.92i)T + (-2.98 - 0.313i)T^{2} \) |
| 11 | \( 1 + (-0.0828 - 0.788i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.692 - 4.36i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.83 + 1.18i)T + (6.91 - 15.5i)T^{2} \) |
| 19 | \( 1 + (4.58 - 5.09i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.628 - 0.241i)T + (17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (-2.48 - 0.808i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.60 + 7.53i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (1.02 - 0.833i)T + (7.69 - 36.1i)T^{2} \) |
| 41 | \( 1 + (-4.00 + 5.51i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (8.14 - 8.14i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.84 + 2.84i)T + (-19.1 - 42.9i)T^{2} \) |
| 53 | \( 1 + (-1.74 - 0.0914i)T + (52.7 + 5.54i)T^{2} \) |
| 59 | \( 1 + (-13.0 + 5.79i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.32 - 2.97i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.53 + 2.37i)T + (-27.2 + 61.2i)T^{2} \) |
| 71 | \( 1 + (3.59 - 11.0i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.0657 - 0.0812i)T + (-15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-1.41 - 6.64i)T + (-72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (0.268 - 0.527i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (2.41 + 1.07i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-3.33 - 6.54i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.27298467251065110024362199565, −11.28652989517408312389848790529, −10.19979855069705567722233406121, −9.676721054690453336793385257205, −8.340872118448319142454997206762, −6.97472986731050584723121449295, −6.20135657580845046234293010533, −4.03722127297749083062957347647, −2.27993697143294211531419618784, −1.32334790709751290683673727280,
2.90954720991636523699732875748, 4.97351449667275079091848415456, 5.73653762414324199030111534548, 6.73062341520125533310843604992, 8.390336198971653748554896848454, 8.936095777652975929521133863134, 9.883336064547978894732878539926, 10.71545091436277120777319994275, 12.28785334529608169473504897985, 13.35611413256575743750484248263
