| L(s) = 1 | + 1.41·3-s + 1.41·5-s − 0.999·9-s + 11-s + 2.82·13-s + 2.00·15-s − 5.65·17-s − 2.82·19-s + 2·23-s − 2.99·25-s − 5.65·27-s − 6·29-s + 1.41·31-s + 1.41·33-s − 2·37-s + 4.00·39-s + 5.65·41-s − 4·43-s − 1.41·45-s − 4.24·47-s − 8.00·51-s − 12·53-s + 1.41·55-s − 4.00·57-s − 4.24·59-s − 5.65·61-s + 4.00·65-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.632·5-s − 0.333·9-s + 0.301·11-s + 0.784·13-s + 0.516·15-s − 1.37·17-s − 0.648·19-s + 0.417·23-s − 0.599·25-s − 1.08·27-s − 1.11·29-s + 0.254·31-s + 0.246·33-s − 0.328·37-s + 0.640·39-s + 0.883·41-s − 0.609·43-s − 0.210·45-s − 0.618·47-s − 1.12·51-s − 1.64·53-s + 0.190·55-s − 0.529·57-s − 0.552·59-s − 0.724·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.57876736028286376065700324215, −6.58457650196766177452272450964, −6.19857826851861701516336484992, −5.43056530756312923324311392347, −4.50153256994148336194726783437, −3.78276805804753880596918911825, −3.02267128261271388800815658486, −2.17502743322467007387981342736, −1.58126441160005025325976768153, 0,
1.58126441160005025325976768153, 2.17502743322467007387981342736, 3.02267128261271388800815658486, 3.78276805804753880596918911825, 4.50153256994148336194726783437, 5.43056530756312923324311392347, 6.19857826851861701516336484992, 6.58457650196766177452272450964, 7.57876736028286376065700324215
