| L(s) = 1 | + 2-s + 3-s + 4-s − 2.24·5-s + 6-s + 3.57·7-s + 8-s + 9-s − 2.24·10-s − 3.14·11-s + 12-s + 13-s + 3.57·14-s − 2.24·15-s + 16-s − 2.24·17-s + 18-s − 6.27·19-s − 2.24·20-s + 3.57·21-s − 3.14·22-s − 4.02·23-s + 24-s + 0.0605·25-s + 26-s + 27-s + 3.57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s + 1.35·7-s + 0.353·8-s + 0.333·9-s − 0.711·10-s − 0.948·11-s + 0.288·12-s + 0.277·13-s + 0.954·14-s − 0.580·15-s + 0.250·16-s − 0.543·17-s + 0.235·18-s − 1.44·19-s − 0.503·20-s + 0.779·21-s − 0.670·22-s − 0.840·23-s + 0.204·24-s + 0.0121·25-s + 0.196·26-s + 0.192·27-s + 0.675·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 6.27T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 + 9.07T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 3.20T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 6.22T + 89T^{2} \) |
| 97 | \( 1 - 0.109T + 97T^{2} \) |
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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
−7.79324073433344235156722455877, −6.82691856936608930654667936311, −6.09177914107883105496634513768, −5.11016339678119709355871689114, −4.53468392081105039788963763168, −4.05906350560207088046865283804, −3.22055317338755301849963008454, −2.27580400915777920317464521794, −1.63617391401644762056752192713, 0,
1.63617391401644762056752192713, 2.27580400915777920317464521794, 3.22055317338755301849963008454, 4.05906350560207088046865283804, 4.53468392081105039788963763168, 5.11016339678119709355871689114, 6.09177914107883105496634513768, 6.82691856936608930654667936311, 7.79324073433344235156722455877
