| L(s) = 1 | − 5-s + 4·7-s − 4·11-s + 13-s + 6·17-s + 4·19-s − 8·23-s + 25-s − 29-s − 8·31-s − 4·35-s − 6·37-s + 2·41-s + 4·43-s + 9·49-s + 10·53-s + 4·55-s − 8·59-s − 10·61-s − 65-s + 8·67-s − 4·71-s + 14·73-s − 16·77-s + 16·79-s − 16·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.185·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.312·41-s + 0.609·43-s + 9/7·49-s + 1.37·53-s + 0.539·55-s − 1.04·59-s − 1.28·61-s − 0.124·65-s + 0.977·67-s − 0.474·71-s + 1.63·73-s − 1.82·77-s + 1.80·79-s − 1.75·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.320453006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.320453006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.46248639811768, −12.35330997371298, −11.92532795198686, −11.37969291255362, −10.80173089046091, −10.73393071412101, −10.07154980767818, −9.558469154236148, −9.089107146460321, −8.277530853704496, −8.027732060767326, −7.862997281261081, −7.257910593272031, −6.887588094543397, −5.780522385537400, −5.530330502512339, −5.363396353820962, −4.539699801687741, −4.173368593545174, −3.476813667683055, −3.089986016291792, −2.231057393933717, −1.798421339952703, −1.176602212508807, −0.4272591312140040,
0.4272591312140040, 1.176602212508807, 1.798421339952703, 2.231057393933717, 3.089986016291792, 3.476813667683055, 4.173368593545174, 4.539699801687741, 5.363396353820962, 5.530330502512339, 5.780522385537400, 6.887588094543397, 7.257910593272031, 7.862997281261081, 8.027732060767326, 8.277530853704496, 9.089107146460321, 9.558469154236148, 10.07154980767818, 10.73393071412101, 10.80173089046091, 11.37969291255362, 11.92532795198686, 12.35330997371298, 12.46248639811768
