| L(s) = 1 | + (−0.369 − 1.73i)2-s + (−0.0553 − 0.124i)3-s + (−1.05 + 0.470i)4-s + (2.20 − 0.389i)5-s + (−0.195 + 0.142i)6-s + (1.92 + 1.81i)7-s + (−0.880 − 1.21i)8-s + (1.99 − 2.21i)9-s + (−1.49 − 3.68i)10-s + (−0.599 − 0.665i)11-s + (0.117 + 0.105i)12-s + (−5.62 + 1.82i)13-s + (2.44 − 4.01i)14-s + (−0.170 − 0.252i)15-s + (−3.32 + 3.69i)16-s + (−2.03 + 0.213i)17-s + ⋯ |
| L(s) = 1 | + (−0.261 − 1.22i)2-s + (−0.0319 − 0.0717i)3-s + (−0.528 + 0.235i)4-s + (0.984 − 0.174i)5-s + (−0.0798 + 0.0580i)6-s + (0.726 + 0.687i)7-s + (−0.311 − 0.428i)8-s + (0.664 − 0.738i)9-s + (−0.471 − 1.16i)10-s + (−0.180 − 0.200i)11-s + (0.0337 + 0.0304i)12-s + (−1.56 + 0.506i)13-s + (0.654 − 1.07i)14-s + (−0.0439 − 0.0651i)15-s + (−0.832 + 0.924i)16-s + (−0.492 + 0.0518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.760523 - 0.948624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.760523 - 0.948624i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.20 + 0.389i)T \) |
| 7 | \( 1 + (-1.92 - 1.81i)T \) |
| good | 2 | \( 1 + (0.369 + 1.73i)T + (-1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (0.0553 + 0.124i)T + (-2.00 + 2.22i)T^{2} \) |
| 11 | \( 1 + (0.599 + 0.665i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (5.62 - 1.82i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.03 - 0.213i)T + (16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.36 - 1.49i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.0408 - 0.192i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (1.19 + 0.866i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.168 - 1.59i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (6.48 + 5.84i)T + (3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-1.96 - 6.04i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 11.5iT - 43T^{2} \) |
| 47 | \( 1 + (-11.9 - 1.25i)T + (45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (0.853 + 1.91i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-7.95 - 1.69i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (2.47 - 0.526i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (10.1 - 1.06i)T + (65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (9.11 + 6.62i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (7.43 - 6.69i)T + (7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.376 + 3.58i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (1.57 + 2.17i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (6.48 - 1.37i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 14.2i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.26427529583523584611234236642, −11.56385262066647059042871659854, −10.37745977078675554374439939066, −9.569742967026019734352271990576, −8.946708591893733085159152456757, −7.23342661230712160881631685550, −5.91297757800378087021922965116, −4.55570859636852244491009835830, −2.71203266065311621015379184584, −1.58524638710379568443520924362,
2.29319143997185546169314377863, 4.80449369203295697839069022865, 5.51149276739704961349972363345, 7.19036259698563837028225256664, 7.35326215501670400277566211436, 8.758246272225128670322345354108, 9.955488106580494685235392373886, 10.68524775132705945200652058329, 12.07516427089923521011490105034, 13.43371341645434439769179291384
