| L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s − 2·13-s + 2·14-s + 16-s + 2·17-s + 20-s + 2·23-s + 25-s + 2·26-s − 2·28-s − 4·29-s + 4·31-s − 32-s − 2·34-s − 2·35-s − 2·37-s − 40-s + 4·41-s + 10·43-s − 2·46-s − 6·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.223·20-s + 0.417·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.742·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.338·35-s − 0.328·37-s − 0.158·40-s + 0.624·41-s + 1.52·43-s − 0.294·46-s − 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.35986166353145, −14.81561579885973, −14.34753549192632, −13.69812894846193, −13.12018854289139, −12.65051869040564, −12.11434877432419, −11.58582169160198, −10.85139476577514, −10.43619002492185, −9.849606106474068, −9.387185884353110, −9.053085423317351, −8.262595389949146, −7.688573975844767, −7.126455367566742, −6.599402593728332, −5.943583455862539, −5.486605464580698, −4.676248447617819, −3.901473542638969, −3.060576280710377, −2.624875702767520, −1.769742787159959, −0.9454870238082228, 0,
0.9454870238082228, 1.769742787159959, 2.624875702767520, 3.060576280710377, 3.901473542638969, 4.676248447617819, 5.486605464580698, 5.943583455862539, 6.599402593728332, 7.126455367566742, 7.688573975844767, 8.262595389949146, 9.053085423317351, 9.387185884353110, 9.849606106474068, 10.43619002492185, 10.85139476577514, 11.58582169160198, 12.11434877432419, 12.65051869040564, 13.12018854289139, 13.69812894846193, 14.34753549192632, 14.81561579885973, 15.35986166353145
