L(s) = 1 | − 5-s − 7-s + 11-s + 3·13-s + 7·19-s − 6·23-s − 4·25-s + 9·29-s + 35-s − 3·37-s − 8·41-s + 10·43-s − 3·47-s + 49-s − 6·53-s − 55-s − 7·59-s + 10·61-s − 3·65-s − 3·67-s + 8·71-s − 7·73-s − 77-s + 8·79-s + 6·89-s − 3·91-s − 7·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.832·13-s + 1.60·19-s − 1.25·23-s − 4/5·25-s + 1.67·29-s + 0.169·35-s − 0.493·37-s − 1.24·41-s + 1.52·43-s − 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.134·55-s − 0.911·59-s + 1.28·61-s − 0.372·65-s − 0.366·67-s + 0.949·71-s − 0.819·73-s − 0.113·77-s + 0.900·79-s + 0.635·89-s − 0.314·91-s − 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.722984484\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722984484\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−8.094218955753912900820893639579, −7.54115096363758312164732223769, −6.66471426783283419492030865795, −6.08235398014212428109150400765, −5.30751619671345781656783102977, −4.37895705212101231345236474267, −3.63159420571458132639806667811, −3.01986513323253084767977177337, −1.79544999468354926741942614750, −0.71530512831648966851389175863,
0.71530512831648966851389175863, 1.79544999468354926741942614750, 3.01986513323253084767977177337, 3.63159420571458132639806667811, 4.37895705212101231345236474267, 5.30751619671345781656783102977, 6.08235398014212428109150400765, 6.66471426783283419492030865795, 7.54115096363758312164732223769, 8.094218955753912900820893639579