L(s) = 1 | + (−0.781 − 0.623i)5-s + (0.623 + 0.781i)9-s + (0.211 + 1.87i)13-s − 0.445i·17-s + (0.222 + 0.974i)25-s + (−0.781 − 0.623i)29-s + (0.541 + 0.678i)37-s + (1.33 + 1.33i)41-s − 0.999i·45-s + (0.781 − 0.623i)49-s + (−0.900 + 1.43i)53-s + (1.59 − 0.559i)61-s + (1.00 − 1.59i)65-s + (1.90 + 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯ |
L(s) = 1 | + (−0.781 − 0.623i)5-s + (0.623 + 0.781i)9-s + (0.211 + 1.87i)13-s − 0.445i·17-s + (0.222 + 0.974i)25-s + (−0.781 − 0.623i)29-s + (0.541 + 0.678i)37-s + (1.33 + 1.33i)41-s − 0.999i·45-s + (0.781 − 0.623i)49-s + (−0.900 + 1.43i)53-s + (1.59 − 0.559i)61-s + (1.00 − 1.59i)65-s + (1.90 + 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.036875580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.036875580\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.781 + 0.623i)T \) |
| 29 | \( 1 + (0.781 + 0.623i)T \) |
good | 3 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 11 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 13 | \( 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2} \) |
| 17 | \( 1 + 0.445iT - T^{2} \) |
| 19 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 23 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 31 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 37 | \( 1 + (-0.541 - 0.678i)T + (-0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (-1.33 - 1.33i)T + iT^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 - 1.43i)T + (-0.433 - 0.900i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1.59 + 0.559i)T + (0.781 - 0.623i)T^{2} \) |
| 67 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 71 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-1.90 - 0.433i)T + (0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 83 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 89 | \( 1 + (-1.05 + 1.68i)T + (-0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
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
−9.330950453339572598535624550307, −8.446365777188569968629500470860, −7.72793778981727250505910773604, −7.09884278184248528900780446618, −6.24231501506850554212907137588, −5.08113324575349435818295695005, −4.41271792380512078789620648649, −3.85439692142547285669532824514, −2.41273856180707817060281610102, −1.33564074870340698000494528881,
0.808527199387539425724671641099, 2.47268035570756624707280087695, 3.54198594443047290609996824488, 3.93672083381781090002111122143, 5.21988430550303855016514660077, 6.02678347620048283737958884164, 6.88608234712598415581696774790, 7.60799848381889434301728536154, 8.157120858125783819939897285282, 9.094755600018557613281336230325