| L(s) = 1 | + (0.222 + 2.11i)2-s + (2.27 − 0.483i)3-s + (−2.46 + 0.524i)4-s + (1.59 − 1.57i)5-s + (1.52 + 4.70i)6-s + (−2.64 − 0.0903i)7-s + (−0.345 − 1.06i)8-s + (2.19 − 0.977i)9-s + (3.67 + 3.01i)10-s + (−2.44 − 1.09i)11-s + (−5.36 + 2.38i)12-s + (2.40 + 1.74i)13-s + (−0.396 − 5.61i)14-s + (2.85 − 4.34i)15-s + (−2.44 + 1.08i)16-s + (−0.753 + 0.836i)17-s + ⋯ |
| L(s) = 1 | + (0.157 + 1.49i)2-s + (1.31 − 0.279i)3-s + (−1.23 + 0.262i)4-s + (0.711 − 0.702i)5-s + (0.623 + 1.91i)6-s + (−0.999 − 0.0341i)7-s + (−0.122 − 0.375i)8-s + (0.731 − 0.325i)9-s + (1.16 + 0.953i)10-s + (−0.738 − 0.328i)11-s + (−1.54 + 0.689i)12-s + (0.667 + 0.484i)13-s + (−0.106 − 1.50i)14-s + (0.737 − 1.12i)15-s + (−0.610 + 0.271i)16-s + (−0.182 + 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32066 + 1.11227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32066 + 1.11227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.59 + 1.57i)T \) |
| 7 | \( 1 + (2.64 + 0.0903i)T \) |
| good | 2 | \( 1 + (-0.222 - 2.11i)T + (-1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (-2.27 + 0.483i)T + (2.74 - 1.22i)T^{2} \) |
| 11 | \( 1 + (2.44 + 1.09i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-2.40 - 1.74i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.753 - 0.836i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.59 + 0.764i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.759 + 7.22i)T + (-22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (3.16 - 9.74i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.56 + 3.95i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-4.09 + 1.82i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-3.05 - 2.21i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.46T + 43T^{2} \) |
| 47 | \( 1 + (-3.47 - 3.85i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (2.32 - 0.495i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.0789 + 0.750i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (0.0500 + 0.475i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.29 + 2.55i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (3.56 - 10.9i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (8.88 + 3.95i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-2.83 - 3.15i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (4.30 + 13.2i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.850 - 8.09i)T + (-87.0 + 18.5i)T^{2} \) |
| 97 | \( 1 + (-0.823 + 2.53i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−13.10978558879744186700567358841, −12.87036919957519227906399841501, −10.65213445124430318816312382961, −9.186559308314143721539291886325, −8.762597093470928264504090980612, −7.85759803674698954450973573784, −6.63195412338982765767609125352, −5.80747588419592569320720816212, −4.29107665863325585961319990931, −2.51034080708005090855770504927,
2.24003155295095882070539086595, 3.02013280912935288333517700086, 3.96538898438731414307177658729, 5.97320174589115131103636251609, 7.53474916350295747667165273855, 8.969700915307176180723390851869, 9.789938759909550433374249648058, 10.28331731756899302755136591515, 11.34828992455951926561720706274, 12.73457235039948245892344377951
