L(s) = 1 | + 0.141·2-s − 1.97·4-s + 4.38·5-s + 3.47·7-s − 0.562·8-s + 0.619·10-s − 11-s − 0.619·13-s + 0.492·14-s + 3.88·16-s − 0.200·17-s − 7.65·19-s − 8.67·20-s − 0.141·22-s − 2.85·23-s + 14.1·25-s − 0.0875·26-s − 6.88·28-s + 0.407·29-s − 0.653·31-s + 1.67·32-s − 0.0283·34-s + 15.2·35-s + 4.13·37-s − 1.08·38-s − 2.46·40-s + 0.852·41-s + ⋯ |
L(s) = 1 | + 0.100·2-s − 0.989·4-s + 1.95·5-s + 1.31·7-s − 0.199·8-s + 0.195·10-s − 0.301·11-s − 0.171·13-s + 0.131·14-s + 0.970·16-s − 0.0486·17-s − 1.75·19-s − 1.93·20-s − 0.0301·22-s − 0.596·23-s + 2.83·25-s − 0.0171·26-s − 1.30·28-s + 0.0756·29-s − 0.117·31-s + 0.296·32-s − 0.00486·34-s + 2.57·35-s + 0.680·37-s − 0.175·38-s − 0.389·40-s + 0.133·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.798702821\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.798702821\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 0.141T + 2T^{2} \) |
| 5 | \( 1 - 4.38T + 5T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 13 | \( 1 + 0.619T + 13T^{2} \) |
| 17 | \( 1 + 0.200T + 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 - 0.407T + 29T^{2} \) |
| 31 | \( 1 + 0.653T + 31T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 - 0.852T + 41T^{2} \) |
| 43 | \( 1 + 2.11T + 43T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 - 5.55T + 59T^{2} \) |
| 67 | \( 1 + 0.741T + 67T^{2} \) |
| 71 | \( 1 - 9.73T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 7.90T + 97T^{2} \) |
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
−8.347121713179464236433604136952, −7.44895859733552681076777483573, −6.34957984572025740228918263979, −5.88983203208964214443989063934, −5.08772502063612891211397976057, −4.74076050003327452475771269049, −3.82985127738238425116746826000, −2.40715066173872385388776869684, −1.99311848845027408544204974768, −0.902799172959379663664488450822,
0.902799172959379663664488450822, 1.99311848845027408544204974768, 2.40715066173872385388776869684, 3.82985127738238425116746826000, 4.74076050003327452475771269049, 5.08772502063612891211397976057, 5.88983203208964214443989063934, 6.34957984572025740228918263979, 7.44895859733552681076777483573, 8.347121713179464236433604136952