| L(s) = 1 | − 3-s − 3·7-s − 2·9-s + 4·13-s + 3·17-s − 5·19-s + 3·21-s + 4·23-s + 5·27-s − 5·29-s − 7·31-s + 7·37-s − 4·39-s + 8·41-s + 6·43-s + 8·47-s + 2·49-s − 3·51-s − 9·53-s + 5·57-s + 13·61-s + 6·63-s − 12·67-s − 4·69-s + 3·71-s − 6·73-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s + 1.10·13-s + 0.727·17-s − 1.14·19-s + 0.654·21-s + 0.834·23-s + 0.962·27-s − 0.928·29-s − 1.25·31-s + 1.15·37-s − 0.640·39-s + 1.24·41-s + 0.914·43-s + 1.16·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s + 0.662·57-s + 1.66·61-s + 0.755·63-s − 1.46·67-s − 0.481·69-s + 0.356·71-s − 0.702·73-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119124773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119124773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−14.51243830204792, −14.17350142480862, −13.30631212028274, −12.93586897479825, −12.65766081614657, −12.04996928241661, −11.20199451151797, −11.07015473026084, −10.61347573682968, −9.827199183877387, −9.360665108056307, −8.806900656239382, −8.411285082082144, −7.509636234823291, −7.137404989987237, −6.298109607831703, −5.900319267688757, −5.740422622513872, −4.795324923334954, −4.066138441963468, −3.510622644234674, −2.919965213015498, −2.216148982069004, −1.164228685085923, −0.4293775540454138,
0.4293775540454138, 1.164228685085923, 2.216148982069004, 2.919965213015498, 3.510622644234674, 4.066138441963468, 4.795324923334954, 5.740422622513872, 5.900319267688757, 6.298109607831703, 7.137404989987237, 7.509636234823291, 8.411285082082144, 8.806900656239382, 9.360665108056307, 9.827199183877387, 10.61347573682968, 11.07015473026084, 11.20199451151797, 12.04996928241661, 12.65766081614657, 12.93586897479825, 13.30631212028274, 14.17350142480862, 14.51243830204792
