L(s) = 1 | − 2·4-s + 5-s + 7-s − 2·11-s − 5·13-s + 4·16-s − 17-s + 2·19-s − 2·20-s + 23-s + 25-s − 2·28-s − 8·29-s + 31-s + 35-s − 3·37-s + 7·41-s + 4·44-s + 47-s + 49-s + 10·52-s + 8·53-s − 2·55-s − 7·61-s − 8·64-s − 5·65-s + 16·67-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.38·13-s + 16-s − 0.242·17-s + 0.458·19-s − 0.447·20-s + 0.208·23-s + 1/5·25-s − 0.377·28-s − 1.48·29-s + 0.179·31-s + 0.169·35-s − 0.493·37-s + 1.09·41-s + 0.603·44-s + 0.145·47-s + 1/7·49-s + 1.38·52-s + 1.09·53-s − 0.269·55-s − 0.896·61-s − 64-s − 0.620·65-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239238697\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239238697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.106487158050962269682243926675, −7.60510118066806050686071754365, −6.87791714938848219971937454293, −5.70226556815601305222886081684, −5.27735130938913567985494651166, −4.64296374829682455447246102514, −3.81822570519668162360175033796, −2.78530579677966525992222851443, −1.91627444815744177034429417146, −0.59379948344703753005746355543,
0.59379948344703753005746355543, 1.91627444815744177034429417146, 2.78530579677966525992222851443, 3.81822570519668162360175033796, 4.64296374829682455447246102514, 5.27735130938913567985494651166, 5.70226556815601305222886081684, 6.87791714938848219971937454293, 7.60510118066806050686071754365, 8.106487158050962269682243926675