| L(s) = 1 | + (−0.625 − 1.26i)2-s + (0.394 + 0.470i)3-s + (−1.21 + 1.58i)4-s + (−2.94 + 1.37i)5-s + (0.349 − 0.795i)6-s + (1.11 − 3.07i)7-s + (2.77 + 0.549i)8-s + (0.455 − 2.58i)9-s + (3.58 + 2.87i)10-s + (0.252 + 0.145i)11-s + (−1.22 + 0.0543i)12-s + (−5.78 − 4.05i)13-s + (−4.59 + 0.504i)14-s + (−1.81 − 0.844i)15-s + (−1.04 − 3.86i)16-s + (−0.0239 + 0.0167i)17-s + ⋯ |
| L(s) = 1 | + (−0.442 − 0.896i)2-s + (0.227 + 0.271i)3-s + (−0.608 + 0.793i)4-s + (−1.31 + 0.614i)5-s + (0.142 − 0.324i)6-s + (0.422 − 1.16i)7-s + (0.980 + 0.194i)8-s + (0.151 − 0.860i)9-s + (1.13 + 0.909i)10-s + (0.0761 + 0.0439i)11-s + (−0.354 + 0.0156i)12-s + (−1.60 − 1.12i)13-s + (−1.22 + 0.134i)14-s + (−0.467 − 0.217i)15-s + (−0.260 − 0.965i)16-s + (−0.00579 + 0.00405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100100 - 0.501453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100100 - 0.501453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.625 + 1.26i)T \) |
| 37 | \( 1 + (-6.06 + 0.487i)T \) |
| good | 3 | \( 1 + (-0.394 - 0.470i)T + (-0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (2.94 - 1.37i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.11 + 3.07i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.252 - 0.145i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.78 + 4.05i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.0239 - 0.0167i)T + (5.81 - 15.9i)T^{2} \) |
| 19 | \( 1 + (4.69 + 0.410i)T + (18.7 + 3.29i)T^{2} \) |
| 23 | \( 1 + (3.86 + 1.03i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-6.31 + 1.69i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.159 - 0.159i)T + 31iT^{2} \) |
| 41 | \( 1 + (-2.35 + 0.415i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.46 + 6.46i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.89 - 3.40i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.361 - 0.992i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.59 - 1.20i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (1.59 - 2.28i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (-1.04 + 2.85i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-9.28 - 11.0i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 5.54iT - 73T^{2} \) |
| 79 | \( 1 + (8.54 - 3.98i)T + (50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (2.42 - 13.7i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.10 + 8.80i)T + (-57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (0.273 - 1.02i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.27573181523235477965594326943, −10.36249740473142650909650199645, −9.878663139283353224447513088115, −8.388626719841533086115296231784, −7.72732901277241328128694412562, −6.89757619157470705584472767930, −4.56103353873112071026194311572, −3.90623211506812519400209324270, −2.79687114836140157205601924260, −0.42143948422092781201267215188,
2.08581459137258129737231407973, 4.45970521487861500504219594508, 4.99472662518391624977641846219, 6.51647651800021603076343265047, 7.64993126162340375573491888976, 8.215590211198718095537281996354, 8.935372763909567528229362556278, 10.02950479841601410318239068712, 11.41113891547119819037035652460, 12.12779929516401509151980041219
