L(s) = 1 | + 3-s − 1.05·5-s − 4.24·7-s + 9-s + 11-s + 13-s − 1.05·15-s − 4.55·17-s + 0.977·19-s − 4.24·21-s + 2.82·23-s − 3.88·25-s + 27-s + 4.75·29-s + 8.28·31-s + 33-s + 4.47·35-s − 6.14·37-s + 39-s − 6.92·41-s − 7.56·43-s − 1.05·45-s − 0.967·47-s + 11.0·49-s − 4.55·51-s − 5.76·53-s − 1.05·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.471·5-s − 1.60·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.272·15-s − 1.10·17-s + 0.224·19-s − 0.927·21-s + 0.589·23-s − 0.777·25-s + 0.192·27-s + 0.882·29-s + 1.48·31-s + 0.174·33-s + 0.756·35-s − 1.00·37-s + 0.160·39-s − 1.08·41-s − 1.15·43-s − 0.157·45-s − 0.141·47-s + 1.57·49-s − 0.637·51-s − 0.791·53-s − 0.142·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.454810130\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.454810130\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 17 | \( 1 + 4.55T + 17T^{2} \) |
| 19 | \( 1 - 0.977T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 8.28T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 + 0.967T + 47T^{2} \) |
| 53 | \( 1 + 5.76T + 53T^{2} \) |
| 59 | \( 1 - 5.16T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 + 3.74T + 71T^{2} \) |
| 73 | \( 1 - 0.141T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 7.42T + 97T^{2} \) |
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
−8.115245408550987433438514186799, −7.06682787137527901240691635534, −6.69177197386066263626764891323, −6.12074701342741722990695772413, −4.99564301928884440720064819931, −4.20262397782384183817420275541, −3.41902660149345206185323283447, −2.98523536701454257071044742742, −1.93966225224493082027308560893, −0.57846996500935879179241950659,
0.57846996500935879179241950659, 1.93966225224493082027308560893, 2.98523536701454257071044742742, 3.41902660149345206185323283447, 4.20262397782384183817420275541, 4.99564301928884440720064819931, 6.12074701342741722990695772413, 6.69177197386066263626764891323, 7.06682787137527901240691635534, 8.115245408550987433438514186799