| L(s) = 1 | + 1.90·3-s + 0.655·5-s − 2.04·7-s + 0.631·9-s − 1.51·11-s − 1.53·13-s + 1.24·15-s + 0.126·17-s + 4.09·19-s − 3.90·21-s − 7.24·23-s − 4.56·25-s − 4.51·27-s + 8.20·29-s − 1.17·31-s − 2.88·33-s − 1.34·35-s + 7.34·37-s − 2.92·39-s − 4.61·41-s − 8.19·43-s + 0.414·45-s − 7.49·47-s − 2.79·49-s + 0.241·51-s + 11.1·53-s − 0.991·55-s + ⋯ |
| L(s) = 1 | + 1.10·3-s + 0.293·5-s − 0.774·7-s + 0.210·9-s − 0.455·11-s − 0.426·13-s + 0.322·15-s + 0.0307·17-s + 0.940·19-s − 0.852·21-s − 1.51·23-s − 0.913·25-s − 0.868·27-s + 1.52·29-s − 0.211·31-s − 0.501·33-s − 0.227·35-s + 1.20·37-s − 0.469·39-s − 0.720·41-s − 1.24·43-s + 0.0617·45-s − 1.09·47-s − 0.399·49-s + 0.0338·51-s + 1.52·53-s − 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 - 0.655T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 0.126T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 + 3.67T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 4.60T + 79T^{2} \) |
| 83 | \( 1 - 0.483T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.978422024299186383889367474457, −7.57549616880312933606456568005, −6.52932595948690766921692985635, −5.91653937055318823780042973623, −5.01246086403558462218525143481, −4.01761065430738885511917874719, −3.17282462075535079331679933957, −2.63848649905396219931418193062, −1.67886325584550296805181413997, 0,
1.67886325584550296805181413997, 2.63848649905396219931418193062, 3.17282462075535079331679933957, 4.01761065430738885511917874719, 5.01246086403558462218525143481, 5.91653937055318823780042973623, 6.52932595948690766921692985635, 7.57549616880312933606456568005, 7.978422024299186383889367474457
