| L(s) = 1 | − 10.1i·5-s + 33.8·7-s − 13.5i·11-s + 45.9i·13-s − 131.·17-s + 7.89i·19-s + 28.2·23-s + 21.9·25-s − 238. i·29-s + 67.2·31-s − 343. i·35-s − 80.2i·37-s − 299.·41-s − 378. i·43-s + 367.·47-s + ⋯ |
| L(s) = 1 | − 0.907i·5-s + 1.82·7-s − 0.372i·11-s + 0.979i·13-s − 1.87·17-s + 0.0953i·19-s + 0.256·23-s + 0.175·25-s − 1.52i·29-s + 0.389·31-s − 1.65i·35-s − 0.356i·37-s − 1.14·41-s − 1.34i·43-s + 1.14·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.158311958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.158311958\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 10.1iT - 125T^{2} \) |
| 7 | \( 1 - 33.8T + 343T^{2} \) |
| 11 | \( 1 + 13.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 45.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.89iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 28.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 238. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 80.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 378. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 28.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 597. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 142. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 656. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 982.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 900.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 650. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 25.6T + 9.12e5T^{2} \) |
| 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
−8.768545299696746154148744627582, −8.134003291417722497718216353619, −7.24773320699532540066377091765, −6.29273119331774496018888438235, −5.22176243785919245618925956780, −4.59898804936849675870929108573, −4.06325223685046778729690327420, −2.28804006098053605612196281388, −1.63152340556016789331061654872, −0.46008143415601939185099799536,
1.20591785475206306241392685564, 2.22201007263734129566280385976, 3.09524987671845484827240943074, 4.45351319349287926194622091361, 4.92025075951299421401607192125, 5.97581938810221234524063100750, 7.02005651901997088523371510108, 7.47640627601462899628704088851, 8.492684057146226165533443763280, 8.895939472447611177639556701714
