| L(s) = 1 | − 2·2-s + 4·4-s − 18·5-s + 7·7-s − 8·8-s + 36·10-s + 11·11-s + 56·13-s − 14·14-s + 16·16-s − 36·17-s − 28·19-s − 72·20-s − 22·22-s − 180·23-s + 199·25-s − 112·26-s + 28·28-s + 54·29-s − 334·31-s − 32·32-s + 72·34-s − 126·35-s + 386·37-s + 56·38-s + 144·40-s + 444·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.60·5-s + 0.377·7-s − 0.353·8-s + 1.13·10-s + 0.301·11-s + 1.19·13-s − 0.267·14-s + 1/4·16-s − 0.513·17-s − 0.338·19-s − 0.804·20-s − 0.213·22-s − 1.63·23-s + 1.59·25-s − 0.844·26-s + 0.188·28-s + 0.345·29-s − 1.93·31-s − 0.176·32-s + 0.363·34-s − 0.608·35-s + 1.71·37-s + 0.239·38-s + 0.569·40-s + 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 36 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 180 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 334 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 - 444 T + p^{3} T^{2} \) |
| 43 | \( 1 + 316 T + p^{3} T^{2} \) |
| 47 | \( 1 - 402 T + p^{3} T^{2} \) |
| 53 | \( 1 - 486 T + p^{3} T^{2} \) |
| 59 | \( 1 - 282 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 - 176 T + p^{3} T^{2} \) |
| 71 | \( 1 - 324 T + p^{3} T^{2} \) |
| 73 | \( 1 - 800 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1144 T + p^{3} T^{2} \) |
| 83 | \( 1 + 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−8.601958322180537194457897430896, −8.093757181429723510854923906201, −7.43940937037103612952648823025, −6.56181679555120029678857060256, −5.60202846449636198930935151356, −4.06505328134045114802122915177, −3.89503016056136020739562413085, −2.39520235077224935073664593844, −1.07223753027673897970923987892, 0,
1.07223753027673897970923987892, 2.39520235077224935073664593844, 3.89503016056136020739562413085, 4.06505328134045114802122915177, 5.60202846449636198930935151356, 6.56181679555120029678857060256, 7.43940937037103612952648823025, 8.093757181429723510854923906201, 8.601958322180537194457897430896
