| L(s) = 1 | + 2·2-s − 8.31·3-s + 4·4-s − 11.7·5-s − 16.6·6-s − 28.8·7-s + 8·8-s + 42.1·9-s − 23.5·10-s − 9.34·11-s − 33.2·12-s − 87.1·13-s − 57.6·14-s + 97.7·15-s + 16·16-s − 3.68·17-s + 84.3·18-s − 159.·19-s − 47.0·20-s + 239.·21-s − 18.6·22-s − 158.·23-s − 66.5·24-s + 13.1·25-s − 174.·26-s − 126.·27-s − 115.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.60·3-s + 0.5·4-s − 1.05·5-s − 1.13·6-s − 1.55·7-s + 0.353·8-s + 1.56·9-s − 0.743·10-s − 0.256·11-s − 0.800·12-s − 1.85·13-s − 1.10·14-s + 1.68·15-s + 0.250·16-s − 0.0525·17-s + 1.10·18-s − 1.92·19-s − 0.525·20-s + 2.49·21-s − 0.181·22-s − 1.44·23-s − 0.565·24-s + 0.105·25-s − 1.31·26-s − 0.899·27-s − 0.778·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 626 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06232169246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06232169246\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 313 | \( 1 - 313T \) |
| good | 3 | \( 1 + 8.31T + 27T^{2} \) |
| 5 | \( 1 + 11.7T + 125T^{2} \) |
| 7 | \( 1 + 28.8T + 343T^{2} \) |
| 11 | \( 1 + 9.34T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.68T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 63.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 190.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 396.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 13.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 843.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 27.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 613.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 680.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 567.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 882.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 352.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−10.26633297061397150317073076712, −9.889154426334932144609768207722, −8.194101646629008829719656206438, −7.02820091720839225534856098175, −6.56293777194321022478190071021, −5.66797573239180943789246240110, −4.62456074847031143257366519888, −3.91149951182477957259191599823, −2.49251329771291302302021460649, −0.13196424200891174924754400075,
0.13196424200891174924754400075, 2.49251329771291302302021460649, 3.91149951182477957259191599823, 4.62456074847031143257366519888, 5.66797573239180943789246240110, 6.56293777194321022478190071021, 7.02820091720839225534856098175, 8.194101646629008829719656206438, 9.889154426334932144609768207722, 10.26633297061397150317073076712
