| L(s) = 1 | − 9.91·3-s + 8.20·5-s + 107.·7-s − 144.·9-s + 167.·11-s − 534.·13-s − 81.3·15-s + 2.01e3·17-s − 1.85e3·19-s − 1.06e3·21-s + 1.09e3·23-s − 3.05e3·25-s + 3.84e3·27-s − 609.·29-s + 3.02e3·31-s − 1.66e3·33-s + 884.·35-s − 4.67e3·37-s + 5.29e3·39-s + 526.·41-s + 6.56e3·43-s − 1.18e3·45-s + 1.15e4·47-s − 5.18e3·49-s − 1.99e4·51-s − 1.57e4·53-s + 1.37e3·55-s + ⋯ |
| L(s) = 1 | − 0.636·3-s + 0.146·5-s + 0.831·7-s − 0.595·9-s + 0.417·11-s − 0.876·13-s − 0.0933·15-s + 1.69·17-s − 1.18·19-s − 0.529·21-s + 0.432·23-s − 0.978·25-s + 1.01·27-s − 0.134·29-s + 0.564·31-s − 0.265·33-s + 0.122·35-s − 0.560·37-s + 0.557·39-s + 0.0488·41-s + 0.541·43-s − 0.0873·45-s + 0.763·47-s − 0.308·49-s − 1.07·51-s − 0.769·53-s + 0.0613·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.626064618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626064618\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 + 6.60e4T \) |
| good | 3 | \( 1 + 9.91T + 243T^{2} \) |
| 5 | \( 1 - 8.20T + 3.12e3T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 167.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 534.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.85e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 609.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 526.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.40e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.36e4T + 8.58e9T^{2} \) |
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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
−9.233502880449669872684588991912, −8.287463145425218019642718470804, −7.61536775095577627716277843860, −6.56573058925420265874625803600, −5.67542406086753855615241225154, −5.06830809955061546749172120298, −4.06423139134252062956837685737, −2.81343085225842032088452352820, −1.69554903777513702292235231922, −0.57832840953691345272646089971,
0.57832840953691345272646089971, 1.69554903777513702292235231922, 2.81343085225842032088452352820, 4.06423139134252062956837685737, 5.06830809955061546749172120298, 5.67542406086753855615241225154, 6.56573058925420265874625803600, 7.61536775095577627716277843860, 8.287463145425218019642718470804, 9.233502880449669872684588991912
