| L(s) = 1 | − 2-s + 4-s − 8-s − 5·13-s + 16-s + 3·17-s − 2·19-s + 9·23-s − 5·25-s + 5·26-s + 3·29-s − 5·31-s − 32-s − 3·34-s + 2·37-s + 2·38-s − 6·41-s − 43-s − 9·46-s − 6·47-s + 5·50-s − 5·52-s − 3·53-s − 3·58-s − 3·59-s + 10·61-s + 5·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.38·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 1.87·23-s − 25-s + 0.980·26-s + 0.557·29-s − 0.898·31-s − 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s − 1.32·46-s − 0.875·47-s + 0.707·50-s − 0.693·52-s − 0.412·53-s − 0.393·58-s − 0.390·59-s + 1.28·61-s + 0.635·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.574049298258472467392444731185, −7.62887873867370361852526319092, −7.21285497786280286801180495079, −6.33738862277121156447234649392, −5.36650684226892599309211492342, −4.64577805751899849665319383092, −3.38877464404185294878323159152, −2.54401642319011399557920759889, −1.43423414088871078406280124121, 0,
1.43423414088871078406280124121, 2.54401642319011399557920759889, 3.38877464404185294878323159152, 4.64577805751899849665319383092, 5.36650684226892599309211492342, 6.33738862277121156447234649392, 7.21285497786280286801180495079, 7.62887873867370361852526319092, 8.574049298258472467392444731185
