| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 7-s − 8-s + 9-s − 10-s − 2·12-s − 2·13-s + 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·19-s + 20-s + 2·21-s − 6·23-s + 2·24-s + 25-s + 2·26-s + 4·27-s − 28-s + 2·30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s − 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.365·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.23461220841344804375630229756, −6.72807526140070678104993913044, −6.06122833225232613588712744256, −5.70662153866554631036371546781, −4.67970942416430370206510172718, −4.10298937122130449113296812869, −2.69846492159539000676340309496, −2.18762556107558741801026877175, −0.898554668080887064889387725326, 0,
0.898554668080887064889387725326, 2.18762556107558741801026877175, 2.69846492159539000676340309496, 4.10298937122130449113296812869, 4.67970942416430370206510172718, 5.70662153866554631036371546781, 6.06122833225232613588712744256, 6.72807526140070678104993913044, 7.23461220841344804375630229756
