L(s) = 1 | − 1.61·2-s − 1.61·3-s + 1.61·4-s + 0.618·5-s + 2.61·6-s + 7-s − 8-s + 1.61·9-s − 1.00·10-s − 2.61·12-s − 1.61·14-s − 1.00·15-s + 17-s − 2.61·18-s + 1.00·20-s − 1.61·21-s + 1.61·24-s − 0.618·25-s − 27-s + 1.61·28-s + 1.61·30-s + 0.618·31-s + 32-s − 1.61·34-s + 0.618·35-s + 2.61·36-s − 0.618·40-s + ⋯ |
L(s) = 1 | − 1.61·2-s − 1.61·3-s + 1.61·4-s + 0.618·5-s + 2.61·6-s + 7-s − 8-s + 1.61·9-s − 1.00·10-s − 2.61·12-s − 1.61·14-s − 1.00·15-s + 17-s − 2.61·18-s + 1.00·20-s − 1.61·21-s + 1.61·24-s − 0.618·25-s − 27-s + 1.61·28-s + 1.61·30-s + 0.618·31-s + 32-s − 1.61·34-s + 0.618·35-s + 2.61·36-s − 0.618·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2343847461\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2343847461\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−13.65194338558675342126772884310, −12.04645147468382634720510609770, −11.44893583765498948948023432146, −10.44337523099519291188430246284, −9.878008452330803346119504481607, −8.439959401615032609354256443612, −7.30241031392590951611595560944, −6.12094459230689769919386566111, −4.97597257629778139191185706419, −1.52402899467920246594218058938,
1.52402899467920246594218058938, 4.97597257629778139191185706419, 6.12094459230689769919386566111, 7.30241031392590951611595560944, 8.439959401615032609354256443612, 9.878008452330803346119504481607, 10.44337523099519291188430246284, 11.44893583765498948948023432146, 12.04645147468382634720510609770, 13.65194338558675342126772884310