| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s + 13-s + 15-s − 17-s − 21-s − 23-s + 25-s + 27-s − 33-s − 35-s − 37-s + 39-s − 41-s + 45-s − 51-s − 53-s − 55-s + 2·59-s − 61-s − 63-s + 65-s + 2·67-s − 69-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s + 13-s + 15-s − 17-s − 21-s − 23-s + 25-s + 27-s − 33-s − 35-s − 37-s + 39-s − 41-s + 45-s − 51-s − 53-s − 55-s + 2·59-s − 61-s − 63-s + 65-s + 2·67-s − 69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.373512862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373512862\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.19260980509239076389115833106, −9.721400366742269413647765018980, −8.807292923041514421843744068884, −8.192305477960500833928622295629, −6.93474393578243008538082252137, −6.28772661967044870940123184783, −5.18253183070206301185928134390, −3.85750901483639670229666836381, −2.87044116647691062621589106057, −1.89260733917123567951299870818,
1.89260733917123567951299870818, 2.87044116647691062621589106057, 3.85750901483639670229666836381, 5.18253183070206301185928134390, 6.28772661967044870940123184783, 6.93474393578243008538082252137, 8.192305477960500833928622295629, 8.807292923041514421843744068884, 9.721400366742269413647765018980, 10.19260980509239076389115833106
