L(s) = 1 | − 3-s − 4·5-s − 2·7-s + 9-s + 2·13-s + 4·15-s + 4·17-s + 6·19-s + 2·21-s + 11·25-s − 27-s + 10·29-s + 4·31-s + 8·35-s + 2·37-s − 2·39-s − 6·41-s + 6·43-s − 4·45-s + 8·47-s − 3·49-s − 4·51-s − 8·53-s − 6·57-s − 4·59-s + 2·61-s − 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s + 0.970·17-s + 1.37·19-s + 0.436·21-s + 11/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.914·43-s − 0.596·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s − 1.09·53-s − 0.794·57-s − 0.520·59-s + 0.256·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.086140342\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086140342\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.01050293422491, −15.77880852555966, −15.62961962446685, −14.62576601879977, −14.10846946622058, −13.39986884975460, −12.63605442601823, −12.21101807672371, −11.71627504146945, −11.41703361911143, −10.47839158590016, −10.19182897851119, −9.309129197675084, −8.644829149168565, −7.882137790808524, −7.571995508768428, −6.811647322136453, −6.252676616614727, −5.444942112768036, −4.654953159294805, −4.111337228489088, −3.205850335839674, −3.013018497094497, −1.208699027395361, −0.5727725240081691,
0.5727725240081691, 1.208699027395361, 3.013018497094497, 3.205850335839674, 4.111337228489088, 4.654953159294805, 5.444942112768036, 6.252676616614727, 6.811647322136453, 7.571995508768428, 7.882137790808524, 8.644829149168565, 9.309129197675084, 10.19182897851119, 10.47839158590016, 11.41703361911143, 11.71627504146945, 12.21101807672371, 12.63605442601823, 13.39986884975460, 14.10846946622058, 14.62576601879977, 15.62961962446685, 15.77880852555966, 16.01050293422491