| L(s) = 1 | − 4·5-s + 6·13-s + 8·17-s + 11·25-s + 4·29-s + 2·37-s − 8·41-s − 7·49-s + 4·53-s + 10·61-s − 24·65-s + 6·73-s − 32·85-s + 16·89-s − 18·97-s + 20·101-s + 6·109-s + 16·113-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.66·13-s + 1.94·17-s + 11/5·25-s + 0.742·29-s + 0.328·37-s − 1.24·41-s − 49-s + 0.549·53-s + 1.28·61-s − 2.97·65-s + 0.702·73-s − 3.47·85-s + 1.69·89-s − 1.82·97-s + 1.99·101-s + 0.574·109-s + 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150273935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150273935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−10.89233403299450937570709814639, −9.973869655575784504500779347821, −8.612772853079119873302234581371, −8.142751074316513789502919252186, −7.33244289349889698794346569795, −6.25541731904712913819044308910, −5.02525715711667990043884728260, −3.81076717999348126429094994331, −3.29164018328411348390302871759, −1.00089337770198902241003598585,
1.00089337770198902241003598585, 3.29164018328411348390302871759, 3.81076717999348126429094994331, 5.02525715711667990043884728260, 6.25541731904712913819044308910, 7.33244289349889698794346569795, 8.142751074316513789502919252186, 8.612772853079119873302234581371, 9.973869655575784504500779347821, 10.89233403299450937570709814639
