| L(s) = 1 | − 0.901·3-s + 2.78·5-s + 2.28·7-s − 2.18·9-s − 0.788·11-s + 5.18·13-s − 2.51·15-s + 4.28·17-s − 3.07·19-s − 2.06·21-s − 23-s + 2.77·25-s + 4.67·27-s + 3.61·29-s − 9.24·31-s + 0.710·33-s + 6.37·35-s + 2.78·37-s − 4.67·39-s − 2.76·41-s + 12.8·43-s − 6.10·45-s + 7.05·47-s − 1.77·49-s − 3.86·51-s + 0.727·53-s − 2.19·55-s + ⋯ |
| L(s) = 1 | − 0.520·3-s + 1.24·5-s + 0.864·7-s − 0.729·9-s − 0.237·11-s + 1.43·13-s − 0.648·15-s + 1.03·17-s − 0.705·19-s − 0.449·21-s − 0.208·23-s + 0.554·25-s + 0.899·27-s + 0.670·29-s − 1.66·31-s + 0.123·33-s + 1.07·35-s + 0.458·37-s − 0.748·39-s − 0.431·41-s + 1.96·43-s − 0.909·45-s + 1.02·47-s − 0.253·49-s − 0.540·51-s + 0.0999·53-s − 0.296·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728588327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728588327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.901T + 3T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + 0.788T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 - 0.727T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 3.78T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 - 8.56T + 73T^{2} \) |
| 79 | \( 1 - 7.95T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−10.68494697867051035014799568467, −9.536470241097514365122252510901, −8.680751122179606526847574209636, −7.939016224829811433644764878456, −6.60891259303916792715896043875, −5.68958670015648366266392232138, −5.41576338883794551050556107639, −3.96245924551469754456285829070, −2.49199369297629938181646538651, −1.26864436567642832599207236628,
1.26864436567642832599207236628, 2.49199369297629938181646538651, 3.96245924551469754456285829070, 5.41576338883794551050556107639, 5.68958670015648366266392232138, 6.60891259303916792715896043875, 7.939016224829811433644764878456, 8.680751122179606526847574209636, 9.536470241097514365122252510901, 10.68494697867051035014799568467
