L(s) = 1 | − 14.0·5-s − 24.2·7-s + 3.10·11-s − 13·13-s − 43.9·17-s + 85.8·19-s + 203.·23-s + 73.4·25-s + 31.0·29-s + 135.·31-s + 341.·35-s + 290.·37-s + 148.·41-s − 281.·43-s − 225.·47-s + 245.·49-s − 172.·53-s − 43.7·55-s − 41.2·59-s + 499.·61-s + 183.·65-s − 503.·67-s + 946.·71-s − 1.11e3·73-s − 75.3·77-s − 674.·79-s + 59.4·83-s + ⋯ |
L(s) = 1 | − 1.25·5-s − 1.31·7-s + 0.0851·11-s − 0.277·13-s − 0.626·17-s + 1.03·19-s + 1.84·23-s + 0.587·25-s + 0.198·29-s + 0.786·31-s + 1.65·35-s + 1.28·37-s + 0.566·41-s − 0.996·43-s − 0.701·47-s + 0.716·49-s − 0.447·53-s − 0.107·55-s − 0.0909·59-s + 1.04·61-s + 0.349·65-s − 0.918·67-s + 1.58·71-s − 1.78·73-s − 0.111·77-s − 0.961·79-s + 0.0786·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
good | 5 | \( 1 + 14.0T + 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 - 3.10T + 1.33e3T^{2} \) |
| 17 | \( 1 + 43.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 85.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 203.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 41.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 503.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 59.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 879.T + 9.12e5T^{2} \) |
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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
−8.468909178207164798214389748841, −7.60624888209001107955132776229, −6.95318381335038903918581270020, −6.29145154370225907360820580309, −5.10490771161957135264713629681, −4.26924507493639001365615677424, −3.32708996039277804782914562409, −2.77219022289939646737603286087, −0.971236304285921129016295712155, 0,
0.971236304285921129016295712155, 2.77219022289939646737603286087, 3.32708996039277804782914562409, 4.26924507493639001365615677424, 5.10490771161957135264713629681, 6.29145154370225907360820580309, 6.95318381335038903918581270020, 7.60624888209001107955132776229, 8.468909178207164798214389748841