L(s) = 1 | − i·3-s − 3i·7-s − 9-s − 2·11-s + 3i·13-s − 6i·17-s − 7·19-s − 3·21-s + 6i·23-s + i·27-s − 2·29-s − 5·31-s + 2i·33-s + 10i·37-s + 3·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.13i·7-s − 0.333·9-s − 0.603·11-s + 0.832i·13-s − 1.45i·17-s − 1.60·19-s − 0.654·21-s + 1.25i·23-s + 0.192i·27-s − 0.371·29-s − 0.898·31-s + 0.348i·33-s + 1.64i·37-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7398453288\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7398453288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 3iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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
−8.251711654258807428969878703982, −7.48864256819948257202563261604, −7.13719852191543241130875719535, −6.39676063706975165436882843172, −5.53724449978106674949920861592, −4.62142884779934403994223876580, −3.99056034914336343200142618524, −2.94736955497131800590406989821, −2.03726676943916773429991317158, −0.987627823977419018255840836011,
0.21998672096031077742442897395, 2.06766597255283751538373498990, 2.57417481469800850481440410396, 3.71503505122769703031839500594, 4.35569055707554968331559678215, 5.39102918851410175269321190964, 5.81477532772943756049781928814, 6.48954271962727643542959951163, 7.61960859333694318278678758499, 8.382180679225378054065618073833