L(s) = 1 | − 5-s − 7-s − 3·11-s + 2·13-s + 2·17-s − 5·19-s − 3·23-s − 4·25-s + 6·29-s − 3·31-s + 35-s − 9·37-s + 3·41-s + 8·43-s − 4·47-s + 49-s + 12·53-s + 3·55-s + 12·59-s + 4·61-s − 2·65-s − 2·67-s + 71-s + 3·77-s − 4·79-s + 6·83-s − 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.554·13-s + 0.485·17-s − 1.14·19-s − 0.625·23-s − 4/5·25-s + 1.11·29-s − 0.538·31-s + 0.169·35-s − 1.47·37-s + 0.468·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 1.64·53-s + 0.404·55-s + 1.56·59-s + 0.512·61-s − 0.248·65-s − 0.244·67-s + 0.118·71-s + 0.341·77-s − 0.450·79-s + 0.658·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.189865185\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189865185\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.220858251567369747720351025593, −7.36826673824308526454828931511, −6.72294780352544673166965391195, −5.87290623867614865610012498158, −5.33417309427618127516073741487, −4.26110230817488576814376855650, −3.74621421231830989388764114361, −2.79109241798921227695560595507, −1.93834041911221566651566608065, −0.55307393407904666798208817107,
0.55307393407904666798208817107, 1.93834041911221566651566608065, 2.79109241798921227695560595507, 3.74621421231830989388764114361, 4.26110230817488576814376855650, 5.33417309427618127516073741487, 5.87290623867614865610012498158, 6.72294780352544673166965391195, 7.36826673824308526454828931511, 8.220858251567369747720351025593