| L(s) = 1 | + 7.65·3-s − 14.6·5-s + 31.5·9-s − 17.3·11-s − 81.5·13-s − 111.·15-s + 78.3·17-s + 11.1·19-s − 133.·23-s + 88.4·25-s + 34.6·27-s − 93.8·29-s − 315.·31-s − 132.·33-s + 179.·37-s − 623.·39-s − 265.·41-s + 6.73·43-s − 460.·45-s − 172.·47-s + 599.·51-s + 375.·53-s + 253.·55-s + 84.9·57-s − 113.·59-s + 359.·61-s + 1.19e3·65-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 1.30·5-s + 1.16·9-s − 0.475·11-s − 1.73·13-s − 1.92·15-s + 1.11·17-s + 0.134·19-s − 1.21·23-s + 0.707·25-s + 0.246·27-s − 0.600·29-s − 1.82·31-s − 0.699·33-s + 0.796·37-s − 2.56·39-s − 1.01·41-s + 0.0238·43-s − 1.52·45-s − 0.534·47-s + 1.64·51-s + 0.974·53-s + 0.621·55-s + 0.197·57-s − 0.251·59-s + 0.754·61-s + 2.27·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 7.65T + 27T^{2} \) |
| 5 | \( 1 + 14.6T + 125T^{2} \) |
| 11 | \( 1 + 17.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 81.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 133.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 315.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.73T + 7.95e4T^{2} \) |
| 47 | \( 1 + 172.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 113.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 359.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 729.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 447.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 10.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 766.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 532.T + 9.12e5T^{2} \) |
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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
−10.14371372968850424517307834631, −9.475119742042054472882350499059, −8.354529761371577832507883438784, −7.68205857795885533659371044634, −7.27498101264139700084354778043, −5.34210341848904896422491369064, −4.06898233998236004684684060232, −3.26755101227126834127051874450, −2.14430894208667924046422698973, 0,
2.14430894208667924046422698973, 3.26755101227126834127051874450, 4.06898233998236004684684060232, 5.34210341848904896422491369064, 7.27498101264139700084354778043, 7.68205857795885533659371044634, 8.354529761371577832507883438784, 9.475119742042054472882350499059, 10.14371372968850424517307834631
