| L(s) = 1 | − 2-s − 0.350·3-s + 4-s − 0.178·5-s + 0.350·6-s − 3.90·7-s − 8-s − 2.87·9-s + 0.178·10-s − 1.52·11-s − 0.350·12-s − 0.315·13-s + 3.90·14-s + 0.0624·15-s + 16-s + 6.70·17-s + 2.87·18-s − 19-s − 0.178·20-s + 1.36·21-s + 1.52·22-s − 3.64·23-s + 0.350·24-s − 4.96·25-s + 0.315·26-s + 2.05·27-s − 3.90·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.202·3-s + 0.5·4-s − 0.0797·5-s + 0.143·6-s − 1.47·7-s − 0.353·8-s − 0.959·9-s + 0.0563·10-s − 0.460·11-s − 0.101·12-s − 0.0874·13-s + 1.04·14-s + 0.0161·15-s + 0.250·16-s + 1.62·17-s + 0.678·18-s − 0.229·19-s − 0.0398·20-s + 0.298·21-s + 0.325·22-s − 0.759·23-s + 0.0715·24-s − 0.993·25-s + 0.0618·26-s + 0.396·27-s − 0.738·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 0.350T + 3T^{2} \) |
| 5 | \( 1 + 0.178T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 13 | \( 1 + 0.315T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 - 3.11T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 - 2.05T + 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 - 5.78T + 73T^{2} \) |
| 79 | \( 1 + 5.62T + 79T^{2} \) |
| 83 | \( 1 - 5.96T + 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 - 5.63T + 97T^{2} \) |
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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
−7.76533682008439401860134800469, −6.65146247278470681950961566053, −6.22118075500722300222193628552, −5.69470812841626748604717502110, −4.76996564577323324025714536833, −3.56801113907900126264122921585, −3.05238024638914152179724343676, −2.33075897667473427730153638521, −0.912140549070592773162440665721, 0,
0.912140549070592773162440665721, 2.33075897667473427730153638521, 3.05238024638914152179724343676, 3.56801113907900126264122921585, 4.76996564577323324025714536833, 5.69470812841626748604717502110, 6.22118075500722300222193628552, 6.65146247278470681950961566053, 7.76533682008439401860134800469
