L(s) = 1 | + (−0.432 + 2.19i)5-s − i·7-s − 0.626·11-s − 5.49i·13-s − 0.896i·17-s + 6.38·19-s + 3.72i·23-s + (−4.62 − 1.89i)25-s − 7.87·29-s − 7.52·31-s + (2.19 + 0.432i)35-s + 6i·37-s − 7.72·41-s + 1.72i·43-s + 5.87i·47-s + ⋯ |
L(s) = 1 | + (−0.193 + 0.981i)5-s − 0.377i·7-s − 0.188·11-s − 1.52i·13-s − 0.217i·17-s + 1.46·19-s + 0.777i·23-s + (−0.925 − 0.379i)25-s − 1.46·29-s − 1.35·31-s + (0.370 + 0.0730i)35-s + 0.986i·37-s − 1.20·41-s + 0.263i·43-s + 0.857i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3254623927\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3254623927\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.432 - 2.19i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 + 5.49iT - 13T^{2} \) |
| 17 | \( 1 + 0.896iT - 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 - 1.72iT - 43T^{2} \) |
| 47 | \( 1 - 5.87iT - 47T^{2} \) |
| 53 | \( 1 + 6.77iT - 53T^{2} \) |
| 59 | \( 1 - 0.593T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.72iT - 73T^{2} \) |
| 79 | \( 1 + 5.67T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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
−8.355685392625906423274920418571, −7.63807096453190810396446148127, −7.34472550952825130255217640338, −6.51815881066626863511157465911, −5.49359989349876050464440758495, −5.21913317846285199985898191320, −3.75413051061583963924056282823, −3.38260032969613950834882027551, −2.54839530890903612329051783681, −1.28776140359962260374255744011,
0.087242946155818872029956470964, 1.49204392923838265294462345417, 2.19712290563677367933835013702, 3.54584351685850532764271751792, 4.11878493890318628898420547786, 5.12018795335621311572320512659, 5.48217931665581726669983496384, 6.46902090109116536360516419813, 7.30363595260395177421662117300, 7.85145919462225967454923561829