| L(s) = 1 | − 2·5-s − 3·9-s − 6·13-s − 25-s − 10·29-s − 2·37-s − 10·41-s + 6·45-s − 7·49-s − 14·53-s − 10·61-s + 12·65-s + 6·73-s + 9·81-s + 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.66·13-s − 1/5·25-s − 1.85·29-s − 0.328·37-s − 1.56·41-s + 0.894·45-s − 49-s − 1.92·53-s − 1.28·61-s + 1.48·65-s + 0.702·73-s + 81-s + 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.41139148306097, −15.68014070354751, −15.02347669870304, −14.90065908513053, −14.17514136725227, −13.69716903836960, −12.88724628026910, −12.35317161202679, −11.91512481602633, −11.32344030410068, −11.00013320576722, −10.11380251625983, −9.564009360242546, −9.034817023594925, −8.281440326899827, −7.743271280419343, −7.376789118727272, −6.594085517114712, −5.888286882148094, −5.082418334932807, −4.757960636394168, −3.741160032473476, −3.267286720059900, −2.452663280139293, −1.657392291607183, 0, 0,
1.657392291607183, 2.452663280139293, 3.267286720059900, 3.741160032473476, 4.757960636394168, 5.082418334932807, 5.888286882148094, 6.594085517114712, 7.376789118727272, 7.743271280419343, 8.281440326899827, 9.034817023594925, 9.564009360242546, 10.11380251625983, 11.00013320576722, 11.32344030410068, 11.91512481602633, 12.35317161202679, 12.88724628026910, 13.69716903836960, 14.17514136725227, 14.90065908513053, 15.02347669870304, 15.68014070354751, 16.41139148306097
