| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 2·13-s + 4·14-s + 15-s + 16-s − 2·17-s − 18-s − 20-s + 4·21-s − 22-s + 24-s + 25-s − 2·26-s − 27-s − 4·28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.872·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.23166652189319, −12.99300517943547, −12.43466727060253, −11.86997713606626, −11.48845517582562, −11.13706387313491, −10.48468750773523, −10.09007153743504, −9.696979607998577, −9.128619729585483, −8.777547012684529, −8.224158952760740, −7.543620138496685, −7.109692076410355, −6.621753282792129, −6.307453129587749, −5.709761752012633, −5.266404385371651, −4.352486228014480, −3.838550606149318, −3.477105774111811, −2.715622595549317, −2.166800234525498, −1.265883207676156, −0.6266537312100570, 0,
0.6266537312100570, 1.265883207676156, 2.166800234525498, 2.715622595549317, 3.477105774111811, 3.838550606149318, 4.352486228014480, 5.266404385371651, 5.709761752012633, 6.307453129587749, 6.621753282792129, 7.109692076410355, 7.543620138496685, 8.224158952760740, 8.777547012684529, 9.128619729585483, 9.696979607998577, 10.09007153743504, 10.48468750773523, 11.13706387313491, 11.48845517582562, 11.86997713606626, 12.43466727060253, 12.99300517943547, 13.23166652189319
