| L(s) = 1 | + 3·3-s − 26.3·7-s + 9·9-s − 20·11-s + 40.3·13-s − 101.·17-s − 29.6·19-s − 79.0·21-s + 22.3·23-s + 27·27-s + 187.·29-s − 293.·31-s − 60·33-s + 1.77·37-s + 121.·39-s + 229.·41-s + 146.·43-s − 538.·47-s + 351.·49-s − 303.·51-s − 432.·53-s − 88.9·57-s + 502.·59-s + 937.·61-s − 237.·63-s − 483.·67-s + 67.0·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.42·7-s + 0.333·9-s − 0.548·11-s + 0.860·13-s − 1.44·17-s − 0.357·19-s − 0.821·21-s + 0.202·23-s + 0.192·27-s + 1.20·29-s − 1.69·31-s − 0.316·33-s + 0.00788·37-s + 0.497·39-s + 0.873·41-s + 0.519·43-s − 1.67·47-s + 1.02·49-s − 0.832·51-s − 1.11·53-s − 0.206·57-s + 1.10·59-s + 1.96·61-s − 0.474·63-s − 0.880·67-s + 0.117·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.614051229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614051229\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 26.3T + 343T^{2} \) |
| 11 | \( 1 + 20T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 293.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.77T + 5.06e4T^{2} \) |
| 41 | \( 1 - 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 538.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 432.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 502.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 937.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 483.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 397.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 56.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 473.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 436.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−8.699111568956411034994691350798, −7.972797923850046425942233639477, −6.87606134743568069931741584798, −6.52074916157754696404911604760, −5.57668644348285890350082468758, −4.44917709277940292422147060346, −3.62210228902047324881771690163, −2.87623413840602983364102840083, −1.97705329061270003761157727461, −0.52223985642771023251088128389,
0.52223985642771023251088128389, 1.97705329061270003761157727461, 2.87623413840602983364102840083, 3.62210228902047324881771690163, 4.44917709277940292422147060346, 5.57668644348285890350082468758, 6.52074916157754696404911604760, 6.87606134743568069931741584798, 7.972797923850046425942233639477, 8.699111568956411034994691350798
