| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 4·5-s + 2·6-s − 7-s + 9-s + 8·10-s − 5·11-s + 2·12-s − 2·13-s − 2·14-s + 4·15-s − 4·16-s + 2·18-s − 5·19-s + 8·20-s − 21-s − 10·22-s − 23-s + 11·25-s − 4·26-s + 27-s − 2·28-s − 2·29-s + 8·30-s + 6·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 1.78·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 2.52·10-s − 1.50·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 1.03·15-s − 16-s + 0.471·18-s − 1.14·19-s + 1.78·20-s − 0.218·21-s − 2.13·22-s − 0.208·23-s + 11/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 1.46·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.767442407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.767442407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.94122298294594700260446625869, −10.06348584847165124277699778695, −9.421863300952081605429638012938, −8.292339792365995708433110594415, −6.91642082849532995462484101470, −6.00450627059691435421291295880, −5.33569718385275672521817298149, −4.33710151569730855874203864959, −2.74710044114612301763782652477, −2.32541152601280590701225190627,
2.32541152601280590701225190627, 2.74710044114612301763782652477, 4.33710151569730855874203864959, 5.33569718385275672521817298149, 6.00450627059691435421291295880, 6.91642082849532995462484101470, 8.292339792365995708433110594415, 9.421863300952081605429638012938, 10.06348584847165124277699778695, 10.94122298294594700260446625869
