L(s) = 1 | + 3·5-s + 7-s − 3·11-s − 13-s + 6·17-s + 4·19-s + 3·23-s + 4·25-s + 3·29-s − 5·31-s + 3·35-s + 2·37-s + 3·41-s + 43-s + 9·47-s − 6·49-s − 6·53-s − 9·55-s + 3·59-s − 13·61-s − 3·65-s + 7·67-s + 12·71-s − 10·73-s − 3·77-s − 11·79-s + 9·83-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.904·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.625·23-s + 4/5·25-s + 0.557·29-s − 0.898·31-s + 0.507·35-s + 0.328·37-s + 0.468·41-s + 0.152·43-s + 1.31·47-s − 6/7·49-s − 0.824·53-s − 1.21·55-s + 0.390·59-s − 1.66·61-s − 0.372·65-s + 0.855·67-s + 1.42·71-s − 1.17·73-s − 0.341·77-s − 1.23·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.230374188\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.230374188\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−9.702090382872232304706168176817, −9.044324062031316603425575486661, −7.910252938778095276305610373978, −7.33863505766548836719037800126, −6.13181657170124013100903903614, −5.45456509339651679693427315165, −4.86009350217895805294255984450, −3.31098434535709278954863783943, −2.37448751408578393336540536078, −1.21023528090684925585240090874,
1.21023528090684925585240090874, 2.37448751408578393336540536078, 3.31098434535709278954863783943, 4.86009350217895805294255984450, 5.45456509339651679693427315165, 6.13181657170124013100903903614, 7.33863505766548836719037800126, 7.910252938778095276305610373978, 9.044324062031316603425575486661, 9.702090382872232304706168176817