| L(s) = 1 | − 4.45·2-s + 11.8·4-s − 5.08·7-s − 17.3·8-s + 58.3·11-s − 21.2·13-s + 22.6·14-s − 17.8·16-s + 68.8·17-s − 40.8·19-s − 259.·22-s + 144.·23-s + 94.5·26-s − 60.3·28-s − 220.·29-s + 291.·31-s + 218.·32-s − 307.·34-s − 260.·37-s + 182.·38-s − 169.·41-s + 438.·43-s + 692.·44-s − 643.·46-s + 255.·47-s − 317.·49-s − 252.·52-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.48·4-s − 0.274·7-s − 0.765·8-s + 1.59·11-s − 0.452·13-s + 0.432·14-s − 0.278·16-s + 0.982·17-s − 0.492·19-s − 2.51·22-s + 1.30·23-s + 0.713·26-s − 0.407·28-s − 1.40·29-s + 1.68·31-s + 1.20·32-s − 1.54·34-s − 1.15·37-s + 0.776·38-s − 0.646·41-s + 1.55·43-s + 2.37·44-s − 2.06·46-s + 0.792·47-s − 0.924·49-s − 0.672·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9206714985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9206714985\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.45T + 8T^{2} \) |
| 7 | \( 1 + 5.08T + 343T^{2} \) |
| 11 | \( 1 - 58.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 255.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 214.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 54.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 758.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 904.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 866.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 206.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 463.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 601.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 229.T + 9.12e5T^{2} \) |
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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
−9.906013994678592822516308708743, −9.177470279607106059911822418442, −8.664125895959126253895820511530, −7.54253803159739043167784171038, −6.91863843122226248528436597804, −5.98934696679403548067066027498, −4.50492500557719034792850418726, −3.18214741212199235923671137288, −1.73977946359141147692596194003, −0.73835726156009355318343411011,
0.73835726156009355318343411011, 1.73977946359141147692596194003, 3.18214741212199235923671137288, 4.50492500557719034792850418726, 5.98934696679403548067066027498, 6.91863843122226248528436597804, 7.54253803159739043167784171038, 8.664125895959126253895820511530, 9.177470279607106059911822418442, 9.906013994678592822516308708743
