L(s) = 1 | − 5-s − 1.44·7-s − 2.44·11-s + 0.449·13-s + 5.44·17-s − 2.44·19-s + 23-s + 25-s − 1.89·29-s + 1.89·31-s + 1.44·35-s − 6.34·37-s − 7.89·41-s + 8.89·43-s + 6.44·47-s − 4.89·49-s − 2.55·53-s + 2.44·55-s + 59-s − 0.449·61-s − 0.449·65-s − 4.55·67-s − 3.89·71-s − 8.44·73-s + 3.55·77-s + 10.5·83-s − 5.44·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.547·7-s − 0.738·11-s + 0.124·13-s + 1.32·17-s − 0.561·19-s + 0.208·23-s + 0.200·25-s − 0.352·29-s + 0.341·31-s + 0.245·35-s − 1.04·37-s − 1.23·41-s + 1.35·43-s + 0.940·47-s − 0.699·49-s − 0.350·53-s + 0.330·55-s + 0.130·59-s − 0.0575·61-s − 0.0557·65-s − 0.555·67-s − 0.462·71-s − 0.988·73-s + 0.404·77-s + 1.15·83-s − 0.591·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.285581226\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.285581226\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 + 6.34T + 37T^{2} \) |
| 41 | \( 1 + 7.89T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - T + 59T^{2} \) |
| 61 | \( 1 + 0.449T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 0.898T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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
−7.73285072262981001995590787379, −7.24504605721799886520174161196, −6.42882046768721362081825343201, −5.71513830262932694941768005105, −5.07442331187307036500545587362, −4.21703479477678943176605880548, −3.40364813383620049691553412771, −2.85476452495387125961506831654, −1.74946996222356115703511041724, −0.55025575709203814693376252882,
0.55025575709203814693376252882, 1.74946996222356115703511041724, 2.85476452495387125961506831654, 3.40364813383620049691553412771, 4.21703479477678943176605880548, 5.07442331187307036500545587362, 5.71513830262932694941768005105, 6.42882046768721362081825343201, 7.24504605721799886520174161196, 7.73285072262981001995590787379