| L(s) = 1 | − 2.03·2-s + 3-s + 2.15·4-s + 1.97·5-s − 2.03·6-s + 1.01·7-s − 0.309·8-s + 9-s − 4.02·10-s + 2.68·11-s + 2.15·12-s − 5.80·13-s − 2.06·14-s + 1.97·15-s − 3.67·16-s − 17-s − 2.03·18-s − 2.41·19-s + 4.24·20-s + 1.01·21-s − 5.48·22-s − 3.48·23-s − 0.309·24-s − 1.10·25-s + 11.8·26-s + 27-s + 2.18·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 0.577·3-s + 1.07·4-s + 0.882·5-s − 0.831·6-s + 0.383·7-s − 0.109·8-s + 0.333·9-s − 1.27·10-s + 0.810·11-s + 0.621·12-s − 1.61·13-s − 0.553·14-s + 0.509·15-s − 0.918·16-s − 0.242·17-s − 0.480·18-s − 0.553·19-s + 0.949·20-s + 0.221·21-s − 1.16·22-s − 0.727·23-s − 0.0631·24-s − 0.220·25-s + 2.32·26-s + 0.192·27-s + 0.413·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 3.15T + 29T^{2} \) |
| 31 | \( 1 - 7.99T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 4.25T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.49T + 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 + 9.40T + 61T^{2} \) |
| 67 | \( 1 + 5.86T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 8.65T + 73T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 3.86T + 97T^{2} \) |
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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
−8.074703179559314562667820208950, −7.74619520779241550531201534320, −6.73656045072902387144454256437, −6.29003416081984545369235318626, −4.96803527955798280777400545923, −4.37425381514609494627377658908, −2.99804490489378202255621128154, −1.99771391023845937699762267681, −1.58668441805333506885846940757, 0,
1.58668441805333506885846940757, 1.99771391023845937699762267681, 2.99804490489378202255621128154, 4.37425381514609494627377658908, 4.96803527955798280777400545923, 6.29003416081984545369235318626, 6.73656045072902387144454256437, 7.74619520779241550531201534320, 8.074703179559314562667820208950
