L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s + 11-s − 13-s + 2·15-s + 4·17-s − 2·19-s + 4·21-s + 6·23-s − 25-s + 27-s + 6·29-s − 6·31-s + 33-s + 8·35-s − 39-s − 6·41-s + 4·43-s + 2·45-s − 12·47-s + 9·49-s + 4·51-s − 4·53-s + 2·55-s − 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.516·15-s + 0.970·17-s − 0.458·19-s + 0.872·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.174·33-s + 1.35·35-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.75·47-s + 9/7·49-s + 0.560·51-s − 0.549·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.589114075\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.589114075\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.461556245360188147146984118112, −8.051388486757397276859070093230, −7.18942523712685973339144091988, −6.42296928010588825026530401947, −5.33384529839533787510423824210, −4.95921820279614520161904352289, −3.93419540314619962871985008003, −2.88315264589909237910926173000, −1.92104020838848350320368564924, −1.25867220212437076392149251523,
1.25867220212437076392149251523, 1.92104020838848350320368564924, 2.88315264589909237910926173000, 3.93419540314619962871985008003, 4.95921820279614520161904352289, 5.33384529839533787510423824210, 6.42296928010588825026530401947, 7.18942523712685973339144091988, 8.051388486757397276859070093230, 8.461556245360188147146984118112