| L(s) = 1 | + 0.323·2-s − 1.75·3-s − 7.89·4-s + 5.91·5-s − 0.569·6-s − 5.14·8-s − 23.9·9-s + 1.91·10-s − 60.8·11-s + 13.8·12-s + 13·13-s − 10.4·15-s + 61.4·16-s + 1.36·17-s − 7.73·18-s + 4.20·19-s − 46.7·20-s − 19.6·22-s − 10.6·23-s + 9.05·24-s − 89.9·25-s + 4.20·26-s + 89.5·27-s + 124.·29-s − 3.37·30-s − 90.9·31-s + 61.0·32-s + ⋯ |
| L(s) = 1 | + 0.114·2-s − 0.338·3-s − 0.986·4-s + 0.529·5-s − 0.0387·6-s − 0.227·8-s − 0.885·9-s + 0.0605·10-s − 1.66·11-s + 0.334·12-s + 0.277·13-s − 0.179·15-s + 0.960·16-s + 0.0194·17-s − 0.101·18-s + 0.0507·19-s − 0.522·20-s − 0.190·22-s − 0.0967·23-s + 0.0769·24-s − 0.719·25-s + 0.0317·26-s + 0.638·27-s + 0.794·29-s − 0.0205·30-s − 0.526·31-s + 0.337·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9165402361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9165402361\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 0.323T + 8T^{2} \) |
| 3 | \( 1 + 1.75T + 27T^{2} \) |
| 5 | \( 1 - 5.91T + 125T^{2} \) |
| 11 | \( 1 + 60.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 1.36T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 10.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.98T + 7.95e4T^{2} \) |
| 47 | \( 1 - 243.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 675.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 87.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 122.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 35.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 811.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 911.T + 9.12e5T^{2} \) |
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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
−10.20105676643556813383068492489, −9.350583332032685143015676405177, −8.415004727453847859788972998780, −7.79001566450512120297219790670, −6.29532715256418426997262208792, −5.45365748047739460146722661382, −4.93205434287457061336837738118, −3.53527649087751659065401979489, −2.37466154066137516251455862969, −0.54517485489772946694852008339,
0.54517485489772946694852008339, 2.37466154066137516251455862969, 3.53527649087751659065401979489, 4.93205434287457061336837738118, 5.45365748047739460146722661382, 6.29532715256418426997262208792, 7.79001566450512120297219790670, 8.415004727453847859788972998780, 9.350583332032685143015676405177, 10.20105676643556813383068492489
