| L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s − 13-s + 15-s − 17-s + 8·19-s − 6·23-s + 25-s − 27-s + 4·31-s + 2·33-s + 2·37-s + 39-s + 4·41-s + 4·43-s − 45-s − 7·49-s + 51-s − 6·53-s + 2·55-s − 8·57-s − 4·59-s + 8·61-s + 65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s + 0.624·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.140·51-s − 0.824·53-s + 0.269·55-s − 1.05·57-s − 0.520·59-s + 1.02·61-s + 0.124·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.54843368736735, −14.29576726919149, −13.65844490456838, −13.12251282671712, −12.63365695919495, −11.97641039093889, −11.75522853096178, −11.19394567244792, −10.65530145964294, −10.04574475511446, −9.649676622977304, −9.127553804615306, −8.222948213419288, −7.893905931618140, −7.370907027304450, −6.846005953672885, −6.056819944071841, −5.696563252647074, −4.975738467986373, −4.542026096246376, −3.855159930165610, −3.130894704560222, −2.551117920372647, −1.633317265528633, −0.8261714541204280, 0,
0.8261714541204280, 1.633317265528633, 2.551117920372647, 3.130894704560222, 3.855159930165610, 4.542026096246376, 4.975738467986373, 5.696563252647074, 6.056819944071841, 6.846005953672885, 7.370907027304450, 7.893905931618140, 8.222948213419288, 9.127553804615306, 9.649676622977304, 10.04574475511446, 10.65530145964294, 11.19394567244792, 11.75522853096178, 11.97641039093889, 12.63365695919495, 13.12251282671712, 13.65844490456838, 14.29576726919149, 14.54843368736735
