| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s + 4·11-s − 2·12-s + 13-s − 4·16-s − 4·17-s + 2·18-s − 7·19-s + 8·22-s + 6·23-s − 5·25-s + 2·26-s − 27-s − 8·29-s − 31-s − 8·32-s − 4·33-s − 8·34-s + 2·36-s + 7·37-s − 14·38-s − 39-s + 41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.277·13-s − 16-s − 0.970·17-s + 0.471·18-s − 1.60·19-s + 1.70·22-s + 1.25·23-s − 25-s + 0.392·26-s − 0.192·27-s − 1.48·29-s − 0.179·31-s − 1.41·32-s − 0.696·33-s − 1.37·34-s + 1/3·36-s + 1.15·37-s − 2.27·38-s − 0.160·39-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.31386948908929023087649036665, −6.65891304369571852518035157900, −6.19200639743874814408139021914, −5.58591142595844489468123162094, −4.68516164355500411321261373969, −4.15082837528659746982830676086, −3.62427390821623158862521523672, −2.48773944570656418291039443303, −1.58268808397569527096005319516, 0,
1.58268808397569527096005319516, 2.48773944570656418291039443303, 3.62427390821623158862521523672, 4.15082837528659746982830676086, 4.68516164355500411321261373969, 5.58591142595844489468123162094, 6.19200639743874814408139021914, 6.65891304369571852518035157900, 7.31386948908929023087649036665
