| L(s) = 1 | + 3-s − 3.56·5-s − 1.56·7-s + 9-s − 11-s + 13-s − 3.56·15-s + 5.12·17-s + 3.12·19-s − 1.56·21-s + 2.43·23-s + 7.68·25-s + 27-s − 10.6·29-s − 10.2·31-s − 33-s + 5.56·35-s − 8.24·37-s + 39-s − 7.56·41-s + 1.56·43-s − 3.56·45-s + 8·47-s − 4.56·49-s + 5.12·51-s + 12.2·53-s + 3.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.59·5-s − 0.590·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.919·15-s + 1.24·17-s + 0.716·19-s − 0.340·21-s + 0.508·23-s + 1.53·25-s + 0.192·27-s − 1.98·29-s − 1.84·31-s − 0.174·33-s + 0.940·35-s − 1.35·37-s + 0.160·39-s − 1.18·41-s + 0.238·43-s − 0.530·45-s + 1.16·47-s − 0.651·49-s + 0.717·51-s + 1.68·53-s + 0.480·55-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
| \(L(1)\) |
\(\approx\) |
\(1.291379168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291379168\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 - 1.56T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 61 | \( 1 - 0.438T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 9.36T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 - 7.36T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.80595779587157437017474957025, −7.32888872849393111733850929143, −7.01801082166743967407435847864, −5.67665550360758217578660859020, −5.18807154584119426473437395965, −3.98598789698427384959486986074, −3.56019664288208301345996751617, −3.11872517600283925009599962741, −1.82419931763589426562517955907, −0.55690572736638455146267846549,
0.55690572736638455146267846549, 1.82419931763589426562517955907, 3.11872517600283925009599962741, 3.56019664288208301345996751617, 3.98598789698427384959486986074, 5.18807154584119426473437395965, 5.67665550360758217578660859020, 7.01801082166743967407435847864, 7.32888872849393111733850929143, 7.80595779587157437017474957025
