| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 9-s + 3·11-s + 4·12-s + 5·13-s − 4·16-s + 3·17-s + 2·18-s − 2·19-s + 6·22-s + 23-s + 10·26-s − 4·27-s − 6·29-s − 31-s − 8·32-s + 6·33-s + 6·34-s + 2·36-s − 4·38-s + 10·39-s + 5·41-s + 43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 1/3·9-s + 0.904·11-s + 1.15·12-s + 1.38·13-s − 16-s + 0.727·17-s + 0.471·18-s − 0.458·19-s + 1.27·22-s + 0.208·23-s + 1.96·26-s − 0.769·27-s − 1.11·29-s − 0.179·31-s − 1.41·32-s + 1.04·33-s + 1.02·34-s + 1/3·36-s − 0.648·38-s + 1.60·39-s + 0.780·41-s + 0.152·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.891344614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.891344614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.681626577481819247955617238840, −8.963964600623637253384654725770, −8.325427603837854543717581153032, −7.26453838817357179950808557028, −6.24275371541535608239026924106, −5.61273181246298351272556326283, −4.31195989409792398114081490514, −3.65095396559314576094472784018, −2.99862137388003117186534072134, −1.71069325460735605547679053519,
1.71069325460735605547679053519, 2.99862137388003117186534072134, 3.65095396559314576094472784018, 4.31195989409792398114081490514, 5.61273181246298351272556326283, 6.24275371541535608239026924106, 7.26453838817357179950808557028, 8.325427603837854543717581153032, 8.963964600623637253384654725770, 9.681626577481819247955617238840
