L(s) = 1 | − 1.83·2-s + 2.36·4-s + 1.70·5-s − 2.50·8-s + 9-s − 3.13·10-s + 2.22·16-s − 1.55·17-s − 1.83·18-s + 4.04·20-s + 1.92·25-s − 1.58·32-s + 2.84·34-s + 2.36·36-s + 0.406·37-s − 4.27·40-s − 0.136·43-s + 1.70·45-s + 1.92·47-s + 49-s − 3.52·50-s + 1.92·61-s + 0.677·64-s − 1.15·67-s − 3.66·68-s − 1.98·71-s − 2.50·72-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 2.36·4-s + 1.70·5-s − 2.50·8-s + 9-s − 3.13·10-s + 2.22·16-s − 1.55·17-s − 1.83·18-s + 4.04·20-s + 1.92·25-s − 1.58·32-s + 2.84·34-s + 2.36·36-s + 0.406·37-s − 4.27·40-s − 0.136·43-s + 1.70·45-s + 1.92·47-s + 49-s − 3.52·50-s + 1.92·61-s + 0.677·64-s − 1.15·67-s − 3.66·68-s − 1.98·71-s − 2.50·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7845098447\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7845098447\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1+O(T) \) |
good | 2 | \( 1 + 1.83T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.70T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.55T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.406T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.136T + T^{2} \) |
| 47 | \( 1 - 1.92T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.92T + T^{2} \) |
| 67 | \( 1 + 1.15T + T^{2} \) |
| 71 | \( 1 + 1.98T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.406T + T^{2} \) |
| 89 | \( 1 + 0.136T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.066287298289164649734590574669, −8.676279513434457503314750475804, −7.53351065237974187408225340750, −6.89280432155855056155877788984, −6.33476885387319236905650417939, −5.52667570730994316383483114513, −4.29560419178449441976106352488, −2.61064308321724618646599665274, −2.04330489544036727690828165291, −1.16160852335317665864603467355,
1.16160852335317665864603467355, 2.04330489544036727690828165291, 2.61064308321724618646599665274, 4.29560419178449441976106352488, 5.52667570730994316383483114513, 6.33476885387319236905650417939, 6.89280432155855056155877788984, 7.53351065237974187408225340750, 8.676279513434457503314750475804, 9.066287298289164649734590574669