| L(s) = 1 | + (1.22 + 1.22i)3-s − i·5-s + 2.44i·7-s + 2.99i·9-s + 4.89·11-s − 2·13-s + (1.22 − 1.22i)15-s − 6i·17-s + 4.89i·19-s + (−2.99 + 2.99i)21-s − 2.44·23-s − 25-s + (−3.67 + 3.67i)27-s − 9.79i·31-s + (5.99 + 5.99i)33-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)3-s − 0.447i·5-s + 0.925i·7-s + 0.999i·9-s + 1.47·11-s − 0.554·13-s + (0.316 − 0.316i)15-s − 1.45i·17-s + 1.12i·19-s + (−0.654 + 0.654i)21-s − 0.510·23-s − 0.200·25-s + (−0.707 + 0.707i)27-s − 1.75i·31-s + (1.04 + 1.04i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43925 + 0.596157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43925 + 0.596157i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.79iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 7.34iT - 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 4.89iT - 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−12.04221196389464327936664572000, −11.52199956625468574143960049595, −9.926897748076442001144584483734, −9.375123936841643347833992673525, −8.588896188842406844999798465121, −7.50229089351175350291436840500, −5.97291331918668535864685692226, −4.81455572398157720119299431592, −3.67521549888056452578309702761, −2.17575853976528079729760166823,
1.51124405817603374519978299934, 3.22926287592756244948709443959, 4.31662415329425293006140228648, 6.37378796084654952220164609089, 6.93695740277744433637739623266, 7.993728322752909466097490185745, 9.023836039567602619593387196662, 10.00821849595029717031468043469, 11.10912735253792462343126601525, 12.12923503844817537246235427186
