L(s) = 1 | + 2.53·3-s + 2.53·7-s + 3.42·9-s + 2.95·11-s + 5.48·13-s − 17-s + 5.06·19-s + 6.42·21-s + 7.37·23-s + 1.06·27-s − 7.91·29-s − 1.88·31-s + 7.48·33-s − 3.06·37-s + 13.9·39-s + 11.0·41-s − 9.48·43-s + 0.421·47-s − 0.578·49-s − 2.53·51-s − 6.84·53-s + 12.8·57-s − 1.91·59-s + 8.13·61-s + 8.67·63-s − 7.57·67-s + 18.6·69-s + ⋯ |
L(s) = 1 | + 1.46·3-s + 0.957·7-s + 1.14·9-s + 0.891·11-s + 1.52·13-s − 0.242·17-s + 1.16·19-s + 1.40·21-s + 1.53·23-s + 0.205·27-s − 1.46·29-s − 0.338·31-s + 1.30·33-s − 0.504·37-s + 2.22·39-s + 1.72·41-s − 1.44·43-s + 0.0614·47-s − 0.0826·49-s − 0.354·51-s − 0.939·53-s + 1.70·57-s − 0.248·59-s + 1.04·61-s + 1.09·63-s − 0.925·67-s + 2.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.892728818\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.892728818\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 + 1.88T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 0.421T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 - 8.13T + 61T^{2} \) |
| 67 | \( 1 + 7.57T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 - 6.22T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 2.55T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.096018735241796082376091550529, −7.43770218246972679007733701040, −6.81657015374893964412416508904, −5.83284809715811596074640856684, −5.05022988501516622220133507424, −4.08640900179253426870548850183, −3.54789635270606734688458093918, −2.86328034961926893053626326024, −1.71137595911220364443699303636, −1.24513823759122388134189660716,
1.24513823759122388134189660716, 1.71137595911220364443699303636, 2.86328034961926893053626326024, 3.54789635270606734688458093918, 4.08640900179253426870548850183, 5.05022988501516622220133507424, 5.83284809715811596074640856684, 6.81657015374893964412416508904, 7.43770218246972679007733701040, 8.096018735241796082376091550529