| L(s) = 1 | − 7-s − 3·9-s − 11-s − 2·13-s − 4·17-s + 2·19-s + 5·23-s + 29-s + 2·31-s − 3·37-s + 12·41-s + 11·43-s + 2·47-s + 49-s − 6·53-s + 10·59-s + 4·61-s + 3·63-s + 67-s + 3·71-s + 77-s + 9·79-s + 9·81-s − 2·83-s − 6·89-s + 2·91-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.301·11-s − 0.554·13-s − 0.970·17-s + 0.458·19-s + 1.04·23-s + 0.185·29-s + 0.359·31-s − 0.493·37-s + 1.87·41-s + 1.67·43-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 1.30·59-s + 0.512·61-s + 0.377·63-s + 0.122·67-s + 0.356·71-s + 0.113·77-s + 1.01·79-s + 81-s − 0.219·83-s − 0.635·89-s + 0.209·91-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.307147213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307147213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.946979557810637828110266122124, −8.048590401901644667794517879848, −7.29272145135204733014167985986, −6.52013850667874281088171520219, −5.70311920212082646010122117895, −4.99108820132036350233322834754, −4.04564886560512577161191163729, −2.93642712584085424830348642860, −2.35906746088252823456430877237, −0.68185355263768612545173164265,
0.68185355263768612545173164265, 2.35906746088252823456430877237, 2.93642712584085424830348642860, 4.04564886560512577161191163729, 4.99108820132036350233322834754, 5.70311920212082646010122117895, 6.52013850667874281088171520219, 7.29272145135204733014167985986, 8.048590401901644667794517879848, 8.946979557810637828110266122124
