| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 9-s − 4·11-s + 13-s − 4·15-s − 17-s + 4·19-s − 4·21-s + 8·23-s − 25-s + 4·27-s + 8·29-s + 10·31-s + 8·33-s + 4·35-s + 10·37-s − 2·39-s − 8·41-s + 12·43-s + 2·45-s + 8·47-s − 3·49-s + 2·51-s − 2·53-s − 8·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.03·15-s − 0.242·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s − 1/5·25-s + 0.769·27-s + 1.48·29-s + 1.79·31-s + 1.39·33-s + 0.676·35-s + 1.64·37-s − 0.320·39-s − 1.24·41-s + 1.82·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.280·51-s − 0.274·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.966279446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966279446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.10315793185755, −15.67158510626430, −15.22958364819795, −14.21041833436801, −13.97967739859449, −13.25427686084574, −12.83624139800319, −12.02653601396855, −11.60257236996914, −10.97470913250111, −10.55918288983892, −10.04078562602904, −9.345171431114166, −8.595322651759693, −7.980688734840693, −7.310045749576699, −6.522824105533671, −5.946314827288230, −5.480593552875319, −4.783931060570647, −4.512864858147825, −2.930427233656630, −2.616005459086677, −1.351227863351480, −0.7352805505649561,
0.7352805505649561, 1.351227863351480, 2.616005459086677, 2.930427233656630, 4.512864858147825, 4.783931060570647, 5.480593552875319, 5.946314827288230, 6.522824105533671, 7.310045749576699, 7.980688734840693, 8.595322651759693, 9.345171431114166, 10.04078562602904, 10.55918288983892, 10.97470913250111, 11.60257236996914, 12.02653601396855, 12.83624139800319, 13.25427686084574, 13.97967739859449, 14.21041833436801, 15.22958364819795, 15.67158510626430, 16.10315793185755
