| L(s) = 1 | + 2.67·2-s + 3-s + 5.15·4-s − 0.517·5-s + 2.67·6-s + 4.18·7-s + 8.43·8-s + 9-s − 1.38·10-s + 5.15·12-s − 1.18·13-s + 11.1·14-s − 0.517·15-s + 12.2·16-s − 5.98·17-s + 2.67·18-s + 19-s − 2.66·20-s + 4.18·21-s − 7.08·23-s + 8.43·24-s − 4.73·25-s − 3.18·26-s + 27-s + 21.5·28-s + 9.43·29-s − 1.38·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.57·4-s − 0.231·5-s + 1.09·6-s + 1.58·7-s + 2.98·8-s + 0.333·9-s − 0.438·10-s + 1.48·12-s − 0.329·13-s + 2.98·14-s − 0.133·15-s + 3.06·16-s − 1.45·17-s + 0.630·18-s + 0.229·19-s − 0.596·20-s + 0.912·21-s − 1.47·23-s + 1.72·24-s − 0.946·25-s − 0.623·26-s + 0.192·27-s + 4.07·28-s + 1.75·29-s − 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.34145895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.34145895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 0.517T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 - 5.55T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 + 0.870T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
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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
−7.74376854621051680689200203452, −7.26505524455818593656559379235, −6.26640733791511762048531389492, −5.77729950805090869325748274874, −4.76997896486704103750505964253, −4.33979569449707953800362612326, −4.00481021962613215573252102923, −2.64926741613493434832327778252, −2.34297163451315784393369096676, −1.37620816936933808554376920603,
1.37620816936933808554376920603, 2.34297163451315784393369096676, 2.64926741613493434832327778252, 4.00481021962613215573252102923, 4.33979569449707953800362612326, 4.76997896486704103750505964253, 5.77729950805090869325748274874, 6.26640733791511762048531389492, 7.26505524455818593656559379235, 7.74376854621051680689200203452
