L(s) = 1 | + (−2.12 − 0.707i)5-s − 5.65i·11-s + (−3 + 3i)13-s + 4i·19-s + (2.82 + 2.82i)23-s + (3.99 + 3i)25-s − 1.41·29-s − 8·31-s + (−7 − 7i)37-s − 1.41i·41-s + (−4 + 4i)43-s + (2.82 − 2.82i)47-s − 7i·49-s + (8.48 + 8.48i)53-s + (−4.00 + 12i)55-s + ⋯ |
L(s) = 1 | + (−0.948 − 0.316i)5-s − 1.70i·11-s + (−0.832 + 0.832i)13-s + 0.917i·19-s + (0.589 + 0.589i)23-s + (0.799 + 0.600i)25-s − 0.262·29-s − 1.43·31-s + (−1.15 − 1.15i)37-s − 0.220i·41-s + (−0.609 + 0.609i)43-s + (0.412 − 0.412i)47-s − i·49-s + (1.16 + 1.16i)53-s + (−0.539 + 1.61i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8589648418\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8589648418\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.12 + 0.707i)T \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (4 - 4i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + (-8 - 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (3 - 3i)T - 73iT^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 5i)T + 97iT^{2} \) |
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
−8.724575627253180403404770742728, −8.332897026875798581505699847944, −7.33465587014914282007004129910, −6.88932231472869055656287036115, −5.61103398911146116670702751397, −5.23041157496148776800641566226, −3.81058202995473181300649875159, −3.68210809968150663747679890079, −2.31421799537714128623987791081, −0.937230644602377049920423992920,
0.33868506903208499851164863366, 2.00972853962709022213784459470, 2.93034245293714163263041356078, 3.87820934593951489758875538209, 4.80397789107308835098249926015, 5.24677756765553268985232743078, 6.75039032694179328108388322391, 7.09949788651689873296344335288, 7.73040858505953339814110950137, 8.567233624350497317991145319078