| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 2·13-s + 16-s + 2·17-s − 18-s − 2·19-s + 8·23-s + 24-s − 2·26-s − 27-s − 8·29-s − 4·31-s − 32-s − 2·34-s + 36-s − 6·37-s + 2·38-s − 2·39-s − 10·41-s + 2·43-s − 8·46-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 + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s + 1.66·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 1.48·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.986·37-s + 0.324·38-s − 0.320·39-s − 1.56·41-s + 0.304·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.42159672991465945999332550436, −7.03332737048667142821960752715, −6.22433209622307589802135350707, −5.54415494259279278282156290596, −4.91068125664934290061451749608, −3.82808731971767829328884565947, −3.14320543152521888666730680028, −1.96631726900574192271988152612, −1.16305503249474348751406417943, 0,
1.16305503249474348751406417943, 1.96631726900574192271988152612, 3.14320543152521888666730680028, 3.82808731971767829328884565947, 4.91068125664934290061451749608, 5.54415494259279278282156290596, 6.22433209622307589802135350707, 7.03332737048667142821960752715, 7.42159672991465945999332550436
