| L(s) = 1 | − 3-s − 4·5-s + 9-s + 2·11-s − 13-s + 4·15-s + 6·17-s + 19-s + 23-s + 11·25-s − 27-s − 10·29-s − 7·31-s − 2·33-s + 3·37-s + 39-s + 12·41-s − 7·43-s − 4·45-s − 10·47-s − 6·51-s − 12·53-s − 8·55-s − 57-s − 14·59-s + 10·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 1.03·15-s + 1.45·17-s + 0.229·19-s + 0.208·23-s + 11/5·25-s − 0.192·27-s − 1.85·29-s − 1.25·31-s − 0.348·33-s + 0.493·37-s + 0.160·39-s + 1.87·41-s − 1.06·43-s − 0.596·45-s − 1.45·47-s − 0.840·51-s − 1.64·53-s − 1.07·55-s − 0.132·57-s − 1.82·59-s + 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.84021661137198, −12.66648686669687, −12.32372348077977, −11.67645153415040, −11.45838471449254, −10.95330093799345, −10.78376350857035, −9.777335876927062, −9.571541039090689, −9.055402851987344, −8.320756335570533, −7.878356296209845, −7.481647193217153, −7.253974959409770, −6.551340224366050, −6.002818482294182, −5.432979186545854, −4.896598622915627, −4.431688467203867, −3.801382266495038, −3.433995334394165, −3.071265930265646, −1.975233647019943, −1.333436222182754, −0.6330322803448802, 0,
0.6330322803448802, 1.333436222182754, 1.975233647019943, 3.071265930265646, 3.433995334394165, 3.801382266495038, 4.431688467203867, 4.896598622915627, 5.432979186545854, 6.002818482294182, 6.551340224366050, 7.253974959409770, 7.481647193217153, 7.878356296209845, 8.320756335570533, 9.055402851987344, 9.571541039090689, 9.777335876927062, 10.78376350857035, 10.95330093799345, 11.45838471449254, 11.67645153415040, 12.32372348077977, 12.66648686669687, 12.84021661137198
