| L(s) = 1 | + 3.64·5-s − 7-s − 5.29·11-s − 0.354·17-s + 19-s − 5.29·23-s + 8.29·25-s + 9.64·29-s − 6·31-s − 3.64·35-s + 5.29·37-s + 3.29·41-s − 1.29·43-s + 4.35·47-s + 49-s + 13.6·53-s − 19.2·55-s + 8·59-s + 9.29·61-s + 7.29·67-s + 8.93·71-s − 2·73-s + 5.29·77-s − 3.29·79-s + 14.9·83-s − 1.29·85-s + 3.29·89-s + ⋯ |
| L(s) = 1 | + 1.63·5-s − 0.377·7-s − 1.59·11-s − 0.0859·17-s + 0.229·19-s − 1.10·23-s + 1.65·25-s + 1.79·29-s − 1.07·31-s − 0.616·35-s + 0.869·37-s + 0.514·41-s − 0.196·43-s + 0.635·47-s + 0.142·49-s + 1.87·53-s − 2.60·55-s + 1.04·59-s + 1.18·61-s + 0.890·67-s + 1.06·71-s − 0.234·73-s + 0.603·77-s − 0.370·79-s + 1.63·83-s − 0.140·85-s + 0.348·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.302893185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.302893185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 0.354T + 17T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 9.64T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 3.29T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 9.29T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 - 8.93T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
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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
−8.342498511489194493128300538292, −7.55514568041951029706274017288, −6.70885970927139041592557610147, −5.99358146865372199238851536794, −5.46523503775127360361047023021, −4.83611213004660297604455525197, −3.66354092059792735001229663505, −2.45024797473173932349400087792, −2.29569875313131728817694321872, −0.822697844479424147022668890964,
0.822697844479424147022668890964, 2.29569875313131728817694321872, 2.45024797473173932349400087792, 3.66354092059792735001229663505, 4.83611213004660297604455525197, 5.46523503775127360361047023021, 5.99358146865372199238851536794, 6.70885970927139041592557610147, 7.55514568041951029706274017288, 8.342498511489194493128300538292
