| L(s) = 1 | − 2.79·3-s + 2.30·7-s + 4.82·9-s − 5.01·11-s − 1.36·13-s + 7.56·17-s + 4.96·19-s − 6.44·21-s − 0.225·23-s − 5.09·27-s − 8.18·29-s − 6.87·31-s + 14.0·33-s + 6.20·37-s + 3.81·39-s + 4.69·41-s − 4.30·43-s + 4.39·47-s − 1.69·49-s − 21.1·51-s + 10.4·53-s − 13.8·57-s + 0.0364·59-s + 9.99·61-s + 11.1·63-s − 14.6·67-s + 0.629·69-s + ⋯ |
| L(s) = 1 | − 1.61·3-s + 0.870·7-s + 1.60·9-s − 1.51·11-s − 0.378·13-s + 1.83·17-s + 1.13·19-s − 1.40·21-s − 0.0469·23-s − 0.981·27-s − 1.52·29-s − 1.23·31-s + 2.44·33-s + 1.02·37-s + 0.610·39-s + 0.733·41-s − 0.655·43-s + 0.641·47-s − 0.242·49-s − 2.96·51-s + 1.43·53-s − 1.83·57-s + 0.00474·59-s + 1.27·61-s + 1.39·63-s − 1.79·67-s + 0.0758·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9600297221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9600297221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 5.01T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 0.225T + 23T^{2} \) |
| 29 | \( 1 + 8.18T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 0.0364T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 8.12T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 0.854T + 89T^{2} \) |
| 97 | \( 1 + 5.51T + 97T^{2} \) |
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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
−8.144472193589035773272705788671, −7.52969951757823987134186906716, −7.17913079797107725957699811272, −5.75490366571127792368216143016, −5.52832881642770417602979651425, −5.08509351425393120042852666914, −4.10024356054548065035827679129, −2.94983992876031083564825909209, −1.67617333844947731914245693546, −0.62782474698862431601926461297,
0.62782474698862431601926461297, 1.67617333844947731914245693546, 2.94983992876031083564825909209, 4.10024356054548065035827679129, 5.08509351425393120042852666914, 5.52832881642770417602979651425, 5.75490366571127792368216143016, 7.17913079797107725957699811272, 7.52969951757823987134186906716, 8.144472193589035773272705788671
