| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 2·19-s + 2·21-s − 25-s − 27-s − 6·29-s + 4·35-s − 4·37-s − 2·39-s + 2·41-s + 6·43-s − 2·45-s − 4·47-s − 3·49-s − 6·51-s − 6·53-s + 2·57-s + 4·61-s − 2·63-s − 4·65-s + 10·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.676·35-s − 0.657·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s − 0.298·45-s − 0.583·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s + 0.264·57-s + 0.512·61-s − 0.251·63-s − 0.496·65-s + 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.50829079615868, −16.00971951466519, −15.64491251917748, −14.97803508455140, −14.38742070668832, −13.74234457851948, −12.86863455317934, −12.69943866362257, −11.95912216811279, −11.52101690665867, −10.87388129752083, −10.36967761645660, −9.599622053288786, −9.204499211785280, −8.143044928153716, −7.870448816550289, −7.082662623828338, −6.480110982754820, −5.800436704659364, −5.249569721727065, −4.318159441120237, −3.638210032477148, −3.240512148514403, −2.010749277057079, −0.9336902404482082, 0,
0.9336902404482082, 2.010749277057079, 3.240512148514403, 3.638210032477148, 4.318159441120237, 5.249569721727065, 5.800436704659364, 6.480110982754820, 7.082662623828338, 7.870448816550289, 8.143044928153716, 9.204499211785280, 9.599622053288786, 10.36967761645660, 10.87388129752083, 11.52101690665867, 11.95912216811279, 12.69943866362257, 12.86863455317934, 13.74234457851948, 14.38742070668832, 14.97803508455140, 15.64491251917748, 16.00971951466519, 16.50829079615868
