L(s) = 1 | + 0.414·2-s − 3-s − 1.82·4-s − 3.41·5-s − 0.414·6-s − 1.58·8-s + 9-s − 1.41·10-s − 2·11-s + 1.82·12-s − 2.58·13-s + 3.41·15-s + 3·16-s + 2.24·17-s + 0.414·18-s + 2.82·19-s + 6.24·20-s − 0.828·22-s − 7.65·23-s + 1.58·24-s + 6.65·25-s − 1.07·26-s − 27-s − 6.82·29-s + 1.41·30-s + 1.17·31-s + 4.41·32-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 0.577·3-s − 0.914·4-s − 1.52·5-s − 0.169·6-s − 0.560·8-s + 0.333·9-s − 0.447·10-s − 0.603·11-s + 0.527·12-s − 0.717·13-s + 0.881·15-s + 0.750·16-s + 0.543·17-s + 0.0976·18-s + 0.648·19-s + 1.39·20-s − 0.176·22-s − 1.59·23-s + 0.323·24-s + 1.33·25-s − 0.210·26-s − 0.192·27-s − 1.26·29-s + 0.258·30-s + 0.210·31-s + 0.780·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−12.24862634564251127101582445586, −11.95663633100051336715682572906, −10.62915661160969898296205034709, −9.527103915730750902096128604275, −8.134790574271130183505669457339, −7.42349770612555731170106759369, −5.65388414877058007429182791770, −4.57935404398144771969035979627, −3.53087914044105296455979885058, 0,
3.53087914044105296455979885058, 4.57935404398144771969035979627, 5.65388414877058007429182791770, 7.42349770612555731170106759369, 8.134790574271130183505669457339, 9.527103915730750902096128604275, 10.62915661160969898296205034709, 11.95663633100051336715682572906, 12.24862634564251127101582445586