| L(s) = 1 | + 3-s − 2·9-s + 6·11-s − 5·13-s − 3·17-s + 19-s − 3·23-s − 5·25-s − 5·27-s + 9·29-s − 4·31-s + 6·33-s + 2·37-s − 5·39-s − 8·43-s − 3·51-s − 3·53-s + 57-s + 9·59-s + 10·61-s − 5·67-s − 3·69-s + 6·71-s + 7·73-s − 5·75-s + 10·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.80·11-s − 1.38·13-s − 0.727·17-s + 0.229·19-s − 0.625·23-s − 25-s − 0.962·27-s + 1.67·29-s − 0.718·31-s + 1.04·33-s + 0.328·37-s − 0.800·39-s − 1.21·43-s − 0.420·51-s − 0.412·53-s + 0.132·57-s + 1.17·59-s + 1.28·61-s − 0.610·67-s − 0.361·69-s + 0.712·71-s + 0.819·73-s − 0.577·75-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143775443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143775443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−16.05710859748804, −15.37174194663751, −14.79935216780154, −14.37789233563134, −14.01621114862507, −13.46000282716391, −12.65482892440077, −11.97007488543712, −11.70331043719405, −11.17036498275232, −10.13930753141360, −9.735926858147792, −9.175570415261606, −8.625211646209566, −8.065793249441136, −7.352954754420931, −6.651945142688034, −6.220737471876189, −5.326076063291418, −4.612544359363297, −3.895725658864953, −3.306235687675479, −2.365648774826446, −1.854944444729453, −0.5998812622591777,
0.5998812622591777, 1.854944444729453, 2.365648774826446, 3.306235687675479, 3.895725658864953, 4.612544359363297, 5.326076063291418, 6.220737471876189, 6.651945142688034, 7.352954754420931, 8.065793249441136, 8.625211646209566, 9.175570415261606, 9.735926858147792, 10.13930753141360, 11.17036498275232, 11.70331043719405, 11.97007488543712, 12.65482892440077, 13.46000282716391, 14.01621114862507, 14.37789233563134, 14.79935216780154, 15.37174194663751, 16.05710859748804
