| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s + 6·13-s + 16-s + 17-s + 18-s − 4·19-s + 24-s + 6·26-s + 27-s − 6·29-s + 32-s + 34-s + 36-s + 2·37-s − 4·38-s + 6·39-s + 6·41-s − 4·43-s + 8·47-s + 48-s − 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.204·24-s + 1.17·26-s + 0.192·27-s − 1.11·29-s + 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + 0.960·39-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 0.144·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.94934983448882, −12.67974702865660, −12.04380496660602, −11.48606510605212, −11.02647184582763, −10.83723054716588, −10.18801015398756, −9.711752332865777, −9.072818459903434, −8.813886389113878, −8.150248578116488, −7.885524658651033, −7.318531386326040, −6.672930127771631, −6.306447262086282, −5.875346099282849, −5.342509938613759, −4.730418208075406, −4.116071007014109, −3.788549327298078, −3.359902240689027, −2.707180052714369, −2.165125114113434, −1.525443285223655, −1.051608077478902, 0,
1.051608077478902, 1.525443285223655, 2.165125114113434, 2.707180052714369, 3.359902240689027, 3.788549327298078, 4.116071007014109, 4.730418208075406, 5.342509938613759, 5.875346099282849, 6.306447262086282, 6.672930127771631, 7.318531386326040, 7.885524658651033, 8.150248578116488, 8.813886389113878, 9.072818459903434, 9.711752332865777, 10.18801015398756, 10.83723054716588, 11.02647184582763, 11.48606510605212, 12.04380496660602, 12.67974702865660, 12.94934983448882
