| L(s) = 1 | + 2-s + 1.95·3-s + 4-s + 5-s + 1.95·6-s − 2.77·7-s + 8-s + 0.817·9-s + 10-s − 11-s + 1.95·12-s − 4.27·13-s − 2.77·14-s + 1.95·15-s + 16-s − 5.17·17-s + 0.817·18-s − 3.13·19-s + 20-s − 5.41·21-s − 22-s − 6.13·23-s + 1.95·24-s + 25-s − 4.27·26-s − 4.26·27-s − 2.77·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.12·3-s + 0.5·4-s + 0.447·5-s + 0.797·6-s − 1.04·7-s + 0.353·8-s + 0.272·9-s + 0.316·10-s − 0.301·11-s + 0.563·12-s − 1.18·13-s − 0.740·14-s + 0.504·15-s + 0.250·16-s − 1.25·17-s + 0.192·18-s − 0.719·19-s + 0.223·20-s − 1.18·21-s − 0.213·22-s − 1.27·23-s + 0.398·24-s + 0.200·25-s − 0.838·26-s − 0.820·27-s − 0.523·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 1.95T + 3T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 6.54T + 31T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 4.76T + 53T^{2} \) |
| 59 | \( 1 - 6.75T + 59T^{2} \) |
| 61 | \( 1 - 8.55T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 7.01T + 97T^{2} \) |
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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
−7.998575780812606143493109939864, −7.03728140042517826582319060771, −6.54904726234624899499092335109, −5.77541755866677781339469723496, −4.84199805098046230706668941273, −4.06265715166602358346561906426, −3.24054955714549380750515848063, −2.46202957448895099094674836265, −2.06375705659878414225743145159, 0,
2.06375705659878414225743145159, 2.46202957448895099094674836265, 3.24054955714549380750515848063, 4.06265715166602358346561906426, 4.84199805098046230706668941273, 5.77541755866677781339469723496, 6.54904726234624899499092335109, 7.03728140042517826582319060771, 7.998575780812606143493109939864
