| L(s) = 1 | − 1.00·2-s + 1.28·3-s − 0.991·4-s + 0.810·5-s − 1.28·6-s − 2.46·7-s + 3.00·8-s − 1.35·9-s − 0.813·10-s + 11-s − 1.27·12-s − 4.30·13-s + 2.47·14-s + 1.04·15-s − 1.03·16-s + 2.50·17-s + 1.35·18-s + 7.71·19-s − 0.804·20-s − 3.16·21-s − 1.00·22-s + 7.08·23-s + 3.85·24-s − 4.34·25-s + 4.32·26-s − 5.58·27-s + 2.44·28-s + ⋯ |
| L(s) = 1 | − 0.709·2-s + 0.741·3-s − 0.495·4-s + 0.362·5-s − 0.526·6-s − 0.931·7-s + 1.06·8-s − 0.450·9-s − 0.257·10-s + 0.301·11-s − 0.367·12-s − 1.19·13-s + 0.661·14-s + 0.268·15-s − 0.258·16-s + 0.608·17-s + 0.320·18-s + 1.77·19-s − 0.179·20-s − 0.690·21-s − 0.214·22-s + 1.47·23-s + 0.787·24-s − 0.868·25-s + 0.847·26-s − 1.07·27-s + 0.461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 - 0.810T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 - 2.50T + 17T^{2} \) |
| 19 | \( 1 - 7.71T + 19T^{2} \) |
| 23 | \( 1 - 7.08T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 - 2.93T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 5.96T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.06T + 83T^{2} \) |
| 89 | \( 1 - 4.69T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−9.139478286420025606335424457766, −8.646614263872667401429145768332, −7.41528988006384386685702071423, −7.24013997197385174866907803765, −5.67842653780651080509923427738, −5.07765554143751131824088512084, −3.63396498626556294206620723382, −2.99984526013468952411581102611, −1.62562643660880347502879693687, 0,
1.62562643660880347502879693687, 2.99984526013468952411581102611, 3.63396498626556294206620723382, 5.07765554143751131824088512084, 5.67842653780651080509923427738, 7.24013997197385174866907803765, 7.41528988006384386685702071423, 8.646614263872667401429145768332, 9.139478286420025606335424457766
