| L(s) = 1 | − 160.·2-s + 729·3-s + 1.74e4·4-s + 3.31e3·5-s − 1.16e5·6-s + 4.49e5·7-s − 1.48e6·8-s + 5.31e5·9-s − 5.31e5·10-s + 5.20e6·11-s + 1.27e7·12-s − 3.19e6·13-s − 7.19e7·14-s + 2.41e6·15-s + 9.43e7·16-s + 3.22e7·17-s − 8.50e7·18-s + 2.60e7·19-s + 5.78e7·20-s + 3.27e8·21-s − 8.34e8·22-s + 9.21e8·23-s − 1.08e9·24-s − 1.20e9·25-s + 5.11e8·26-s + 3.87e8·27-s + 7.84e9·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.12·4-s + 0.0949·5-s − 1.02·6-s + 1.44·7-s − 1.99·8-s + 0.333·9-s − 0.168·10-s + 0.886·11-s + 1.22·12-s − 0.183·13-s − 2.55·14-s + 0.0548·15-s + 1.40·16-s + 0.323·17-s − 0.589·18-s + 0.126·19-s + 0.202·20-s + 0.833·21-s − 1.56·22-s + 1.29·23-s − 1.15·24-s − 0.990·25-s + 0.324·26-s + 0.192·27-s + 3.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.8948752790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8948752790\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| good | 2 | \( 1 + 160.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.31e3T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.49e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.20e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.19e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 3.22e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.60e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 9.21e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.17e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.65e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.60e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 5.42e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.58e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.26e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 6.08e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 4.10e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 1.17e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.16e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 2.58e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.87e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.22e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.08e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 1.34e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 3.12e9T + 6.73e25T^{2} \) |
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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
−24.39017838926355392331853791541, −21.08555992258656624404991926653, −19.76871712424685958805129749220, −18.26604968658162857069681661327, −16.94411551325776706406894672490, −14.80330189334061524295227176401, −11.24734511541786311405744334732, −9.209714162808340369177728203170, −7.65399623102471503195496041115, −1.58688330996972788539996400211,
1.58688330996972788539996400211, 7.65399623102471503195496041115, 9.209714162808340369177728203170, 11.24734511541786311405744334732, 14.80330189334061524295227176401, 16.94411551325776706406894672490, 18.26604968658162857069681661327, 19.76871712424685958805129749220, 21.08555992258656624404991926653, 24.39017838926355392331853791541
