L(s) = 1 | + 3-s + 3.69·5-s + 2.24·7-s + 9-s + 5.56·13-s + 3.69·15-s − 1.01·17-s + 2.65·19-s + 2.24·21-s − 2.65·23-s + 8.62·25-s + 27-s − 1.38·29-s − 4.89·31-s + 8.27·35-s − 1.97·37-s + 5.56·39-s + 4.28·41-s − 0.551·43-s + 3.69·45-s + 5.62·47-s − 1.97·49-s − 1.01·51-s + 1.03·53-s + 2.65·57-s + 2.42·59-s + 5.38·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.65·5-s + 0.847·7-s + 0.333·9-s + 1.54·13-s + 0.953·15-s − 0.246·17-s + 0.608·19-s + 0.489·21-s − 0.552·23-s + 1.72·25-s + 0.192·27-s − 0.256·29-s − 0.878·31-s + 1.39·35-s − 0.325·37-s + 0.891·39-s + 0.669·41-s − 0.0840·43-s + 0.550·45-s + 0.820·47-s − 0.281·49-s − 0.142·51-s + 0.142·53-s + 0.351·57-s + 0.316·59-s + 0.689·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.678106528\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.678106528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 + 0.551T + 43T^{2} \) |
| 47 | \( 1 - 5.62T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 - 2.42T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 + 6.27T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.935415353416980516557592632042, −7.14968084155472886833228888867, −6.33201177498050209060347066930, −5.74951495554562112382130368846, −5.20849405937760450944233503296, −4.25249171471864487990168816978, −3.45651291624323972820686779589, −2.48335168892101473527837465214, −1.76242458560598234595187591527, −1.18316245138188585173662989672,
1.18316245138188585173662989672, 1.76242458560598234595187591527, 2.48335168892101473527837465214, 3.45651291624323972820686779589, 4.25249171471864487990168816978, 5.20849405937760450944233503296, 5.74951495554562112382130368846, 6.33201177498050209060347066930, 7.14968084155472886833228888867, 7.935415353416980516557592632042