| L(s) = 1 | + 5-s − 3·11-s + 13-s − 3·17-s − 2·19-s + 6·23-s + 25-s + 9·29-s − 8·31-s − 10·37-s + 2·43-s − 3·47-s − 3·55-s + 12·59-s − 8·61-s + 65-s + 8·67-s − 14·73-s + 5·79-s − 12·83-s − 3·85-s + 12·89-s − 2·95-s − 17·97-s − 6·101-s + 7·103-s + 6·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s + 0.277·13-s − 0.727·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s − 1.43·31-s − 1.64·37-s + 0.304·43-s − 0.437·47-s − 0.404·55-s + 1.56·59-s − 1.02·61-s + 0.124·65-s + 0.977·67-s − 1.63·73-s + 0.562·79-s − 1.31·83-s − 0.325·85-s + 1.27·89-s − 0.205·95-s − 1.72·97-s − 0.597·101-s + 0.689·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.23910517139277563099291689192, −6.81994548162428098844724525508, −6.03887988941668471412937410624, −5.24915802732557648975953248744, −4.80027926903602825537655966957, −3.83399649630466906978434972872, −2.94968504334688159105625821670, −2.26842103556972506213558785442, −1.29413937757275343309010023692, 0,
1.29413937757275343309010023692, 2.26842103556972506213558785442, 2.94968504334688159105625821670, 3.83399649630466906978434972872, 4.80027926903602825537655966957, 5.24915802732557648975953248744, 6.03887988941668471412937410624, 6.81994548162428098844724525508, 7.23910517139277563099291689192
