L(s) = 1 | − 5-s − 3·7-s + 5·13-s − 7·17-s − 7·19-s + 25-s + 7·29-s − 6·31-s + 3·35-s − 5·37-s − 10·41-s + 6·43-s − 10·47-s + 2·49-s + 12·53-s − 12·61-s − 5·65-s + 2·67-s − 9·71-s + 6·73-s − 10·79-s − 13·83-s + 7·85-s − 4·89-s − 15·91-s + 7·95-s + 2·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.38·13-s − 1.69·17-s − 1.60·19-s + 1/5·25-s + 1.29·29-s − 1.07·31-s + 0.507·35-s − 0.821·37-s − 1.56·41-s + 0.914·43-s − 1.45·47-s + 2/7·49-s + 1.64·53-s − 1.53·61-s − 0.620·65-s + 0.244·67-s − 1.06·71-s + 0.702·73-s − 1.12·79-s − 1.42·83-s + 0.759·85-s − 0.423·89-s − 1.57·91-s + 0.718·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08628776456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08628776456\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.71764201719905, −13.28681602870638, −13.04387060994005, −12.50773696963352, −11.97644553199319, −11.33071151901962, −10.94369460455929, −10.42138879732161, −10.11062754686588, −9.178044081865131, −8.857756311287239, −8.506559351315240, −7.994110030600725, −7.019478113402410, −6.719521275721964, −6.376656123660952, −5.811117205560563, −5.043783084758929, −4.248569292606401, −4.036947463132713, −3.314066005077852, −2.775215489378888, −2.001761771378110, −1.301446205505965, −0.09403992298394934,
0.09403992298394934, 1.301446205505965, 2.001761771378110, 2.775215489378888, 3.314066005077852, 4.036947463132713, 4.248569292606401, 5.043783084758929, 5.811117205560563, 6.376656123660952, 6.719521275721964, 7.019478113402410, 7.994110030600725, 8.506559351315240, 8.857756311287239, 9.178044081865131, 10.11062754686588, 10.42138879732161, 10.94369460455929, 11.33071151901962, 11.97644553199319, 12.50773696963352, 13.04387060994005, 13.28681602870638, 13.71764201719905