L(s) = 1 | − 4.24·5-s − 4·7-s − 16.9·11-s + 8i·13-s − 12.7i·17-s + 16i·19-s − 16.9i·23-s − 7.00·25-s − 4.24·29-s − 44·31-s + 16.9·35-s + 34i·37-s − 46.6i·41-s − 40i·43-s + 84.8i·47-s + ⋯ |
L(s) = 1 | − 0.848·5-s − 0.571·7-s − 1.54·11-s + 0.615i·13-s − 0.748i·17-s + 0.842i·19-s − 0.737i·23-s − 0.280·25-s − 0.146·29-s − 1.41·31-s + 0.484·35-s + 0.918i·37-s − 1.13i·41-s − 0.930i·43-s + 1.80i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6990155272\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6990155272\) |
\(L(2)\) |
|
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 | \( 1 + 4.24T + 25T^{2} \) |
| 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 + 16.9T + 121T^{2} \) |
| 13 | \( 1 - 8iT - 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24T + 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 - 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 50iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.790476944314331165706685513588, −7.83396276417147930506094259001, −7.48703228902346282073946365098, −6.54132684023968685569243200869, −5.61362254458356761899676427107, −4.80120543012464786538411172103, −3.87198502247601285257099005230, −3.05763122170834896942337825751, −2.04985678904004030934329213547, −0.37610984496814733943518254230,
0.44524620961814616117590422170, 2.11098884714147171744501206171, 3.17523253079443008466307657615, 3.80460493862554123890398967631, 4.96031454285025270952516475346, 5.59876148754022965635710888658, 6.56973866768123158882179899721, 7.60552203667986504141256040929, 7.83648515653877524939623354292, 8.769033240160922610700971537137