| L(s) = 1 | − 3.69·5-s + (2.41 + 1.08i)7-s − 3.41i·11-s − 5.22i·13-s + 6.75·17-s + 2.16i·19-s − 6.24i·23-s + 8.65·25-s + 2.58i·29-s − 10.4i·31-s + (−8.92 − 4i)35-s + 4·37-s − 0.634·41-s − 6.48·43-s − 3.06·47-s + ⋯ |
| L(s) = 1 | − 1.65·5-s + (0.912 + 0.409i)7-s − 1.02i·11-s − 1.44i·13-s + 1.63·17-s + 0.496i·19-s − 1.30i·23-s + 1.73·25-s + 0.480i·29-s − 1.87i·31-s + (−1.50 − 0.676i)35-s + 0.657·37-s − 0.0990·41-s − 0.988·43-s − 0.446·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.946856 - 0.540220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946856 - 0.540220i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.41 - 1.08i)T \) |
| good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 + 3.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.22iT - 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 - 2.16iT - 19T^{2} \) |
| 23 | \( 1 + 6.24iT - 23T^{2} \) |
| 29 | \( 1 - 2.58iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 0.634T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 7.41iT - 71T^{2} \) |
| 73 | \( 1 - 0.896iT - 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 + 9.55iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96810140477687804952270411387, −10.07724444780262436469299366617, −8.542802439235002399641485595821, −8.065223541814578722308048330273, −7.57360768238681681866818114706, −5.99171796535100730164519734907, −5.07062364885490421084735585311, −3.87055657450852906402591656868, −2.96288243462092114580970615693, −0.74180574762259422332756820605,
1.48625835982229472549645839351, 3.42012462234461023680414576705, 4.34813505461019736265879639520, 5.08174743838834544698850723816, 6.87092016375339358516205447723, 7.50418613245595872773817375380, 8.141253829673798844816250759621, 9.226399338170779321783524921615, 10.27920622127599783463026440530, 11.34321073510283392958202083022
