| L(s) = 1 | − 2-s + 0.988·3-s + 4-s − 5-s − 0.988·6-s + 3.47·7-s − 8-s − 2.02·9-s + 10-s − 6.04·11-s + 0.988·12-s − 3.47·14-s − 0.988·15-s + 16-s + 2.05·17-s + 2.02·18-s − 3.11·19-s − 20-s + 3.43·21-s + 6.04·22-s + 7.97·23-s − 0.988·24-s + 25-s − 4.96·27-s + 3.47·28-s − 6.73·29-s + 0.988·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.570·3-s + 0.5·4-s − 0.447·5-s − 0.403·6-s + 1.31·7-s − 0.353·8-s − 0.674·9-s + 0.316·10-s − 1.82·11-s + 0.285·12-s − 0.928·14-s − 0.255·15-s + 0.250·16-s + 0.498·17-s + 0.476·18-s − 0.715·19-s − 0.223·20-s + 0.749·21-s + 1.28·22-s + 1.66·23-s − 0.201·24-s + 0.200·25-s − 0.955·27-s + 0.656·28-s − 1.24·29-s + 0.180·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.988T + 3T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 2.69T + 37T^{2} \) |
| 41 | \( 1 + 3.23T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 6.40T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 0.121T + 71T^{2} \) |
| 73 | \( 1 - 0.138T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
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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
−8.718021687404665736172267593721, −8.137968518456902273111790899618, −7.75314801448042044901057911522, −6.93247856237820925159953960533, −5.40996477358211761404013690672, −5.10312066496499449838207488787, −3.61381348389768497371453691905, −2.67609557183490871090554064739, −1.74008536247647708732539713003, 0,
1.74008536247647708732539713003, 2.67609557183490871090554064739, 3.61381348389768497371453691905, 5.10312066496499449838207488787, 5.40996477358211761404013690672, 6.93247856237820925159953960533, 7.75314801448042044901057911522, 8.137968518456902273111790899618, 8.718021687404665736172267593721
