| L(s) = 1 | + 4·7-s − 2·13-s + 8·19-s − 5·25-s + 4·31-s + 10·37-s + 8·43-s + 9·49-s − 14·61-s − 16·67-s − 10·73-s + 4·79-s − 8·91-s + 14·97-s − 20·103-s − 2·109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.554·13-s + 1.83·19-s − 25-s + 0.718·31-s + 1.64·37-s + 1.21·43-s + 9/7·49-s − 1.79·61-s − 1.95·67-s − 1.17·73-s + 0.450·79-s − 0.838·91-s + 1.42·97-s − 1.97·103-s − 0.191·109-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
| \(L(1)\) |
\(\approx\) |
\(1.717315342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717315342\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−10.82265776539167631942690570095, −9.825287305444944792800568324534, −9.003535964671978513824781543440, −7.73821550164192930578511356198, −7.59743334158084999547271626311, −6.01970007910918377722950045283, −5.09016984693454778890596036288, −4.25084195754076010426468082779, −2.73087009892762739420775730904, −1.33945102693755199909069874398,
1.33945102693755199909069874398, 2.73087009892762739420775730904, 4.25084195754076010426468082779, 5.09016984693454778890596036288, 6.01970007910918377722950045283, 7.59743334158084999547271626311, 7.73821550164192930578511356198, 9.003535964671978513824781543440, 9.825287305444944792800568324534, 10.82265776539167631942690570095
