L(s) = 1 | − 3.98·2-s + 7.89·4-s − 12.5·7-s + 0.411·8-s + 11·11-s − 36.0·13-s + 50.0·14-s − 64.8·16-s + 39.7·17-s − 148.·19-s − 43.8·22-s − 35.0·23-s + 143.·26-s − 99.2·28-s − 88.2·29-s − 166.·31-s + 255.·32-s − 158.·34-s − 85.2·37-s + 590.·38-s − 329.·41-s + 278.·43-s + 86.8·44-s + 139.·46-s − 272.·47-s − 185.·49-s − 284.·52-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 0.987·4-s − 0.678·7-s + 0.0181·8-s + 0.301·11-s − 0.769·13-s + 0.956·14-s − 1.01·16-s + 0.566·17-s − 1.78·19-s − 0.425·22-s − 0.317·23-s + 1.08·26-s − 0.669·28-s − 0.565·29-s − 0.966·31-s + 1.40·32-s − 0.798·34-s − 0.378·37-s + 2.52·38-s − 1.25·41-s + 0.988·43-s + 0.297·44-s + 0.448·46-s − 0.846·47-s − 0.539·49-s − 0.759·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2864172553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2864172553\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 3.98T + 8T^{2} \) |
| 7 | \( 1 + 12.5T + 343T^{2} \) |
| 13 | \( 1 + 36.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 329.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 467.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 733.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 537.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 397.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 166.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 694.T + 9.12e5T^{2} \) |
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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
−8.723425438074035990666838971945, −7.974350488777853983566986568272, −7.17191769197918802988950665064, −6.62442966988940246820907820104, −5.67357320360548813919007695177, −4.55732724589626237305256342024, −3.62996731052640810240199669027, −2.41154825904253618367974644540, −1.59616287416682399006285397730, −0.29063864191168474430159525186,
0.29063864191168474430159525186, 1.59616287416682399006285397730, 2.41154825904253618367974644540, 3.62996731052640810240199669027, 4.55732724589626237305256342024, 5.67357320360548813919007695177, 6.62442966988940246820907820104, 7.17191769197918802988950665064, 7.974350488777853983566986568272, 8.723425438074035990666838971945