| L(s) = 1 | − 0.211·2-s + 3-s − 1.95·4-s + 5-s − 0.211·6-s + 3.03·7-s + 0.836·8-s + 9-s − 0.211·10-s + 2.42·11-s − 1.95·12-s + 5.84·13-s − 0.641·14-s + 15-s + 3.73·16-s − 0.920·17-s − 0.211·18-s + 2.43·19-s − 1.95·20-s + 3.03·21-s − 0.512·22-s + 0.265·23-s + 0.836·24-s + 25-s − 1.23·26-s + 27-s − 5.93·28-s + ⋯ |
| L(s) = 1 | − 0.149·2-s + 0.577·3-s − 0.977·4-s + 0.447·5-s − 0.0863·6-s + 1.14·7-s + 0.295·8-s + 0.333·9-s − 0.0668·10-s + 0.730·11-s − 0.564·12-s + 1.62·13-s − 0.171·14-s + 0.258·15-s + 0.933·16-s − 0.223·17-s − 0.0498·18-s + 0.557·19-s − 0.437·20-s + 0.662·21-s − 0.109·22-s + 0.0554·23-s + 0.170·24-s + 0.200·25-s − 0.242·26-s + 0.192·27-s − 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.966064704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.966064704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.211T + 2T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 + 0.920T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 0.265T + 23T^{2} \) |
| 29 | \( 1 + 0.369T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 8.25T + 37T^{2} \) |
| 41 | \( 1 + 5.57T + 41T^{2} \) |
| 43 | \( 1 - 6.44T + 43T^{2} \) |
| 47 | \( 1 + 0.0721T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 + 0.801T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 1.32T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 0.184T + 97T^{2} \) |
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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
−8.273197447111868866987036933269, −7.69004002359690633766364328931, −6.63108167091012614380710268669, −5.91352133916692699223341143652, −5.09739354617104367580449004150, −4.34567321735937623332917073009, −3.81279252413477707945992784027, −2.82382959232017067877258244723, −1.54108755071289980136088466358, −1.07498591966058001913181300448,
1.07498591966058001913181300448, 1.54108755071289980136088466358, 2.82382959232017067877258244723, 3.81279252413477707945992784027, 4.34567321735937623332917073009, 5.09739354617104367580449004150, 5.91352133916692699223341143652, 6.63108167091012614380710268669, 7.69004002359690633766364328931, 8.273197447111868866987036933269
