| L(s) = 1 | + 3-s − 2·7-s + 9-s − 13-s + 4·17-s − 6·19-s − 2·21-s + 6·23-s + 27-s + 4·29-s − 8·31-s + 6·37-s − 39-s + 6·41-s + 4·43-s + 8·47-s − 3·49-s + 4·51-s − 2·53-s − 6·57-s − 2·61-s − 2·63-s − 4·67-s + 6·69-s + 8·71-s − 16·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.970·17-s − 1.37·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s + 0.742·29-s − 1.43·31-s + 0.986·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s − 0.794·57-s − 0.256·61-s − 0.251·63-s − 0.488·67-s + 0.722·69-s + 0.949·71-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.272468249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272468249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.05030948641354, −15.31800297857919, −14.88461647192364, −14.39503683217845, −13.86240519599327, −13.08469015358571, −12.62853796793232, −12.46083751452920, −11.43336368092681, −10.83137592976229, −10.34321895800460, −9.573278338766924, −9.257173661936343, −8.579514169017684, −7.957486674793232, −7.262194210605412, −6.802207751494452, −6.002643432898759, −5.451969067429210, −4.470930160098801, −3.982884946057257, −3.056862410700615, −2.670188833334527, −1.666991760331947, −0.6317435945058356,
0.6317435945058356, 1.666991760331947, 2.670188833334527, 3.056862410700615, 3.982884946057257, 4.470930160098801, 5.451969067429210, 6.002643432898759, 6.802207751494452, 7.262194210605412, 7.957486674793232, 8.579514169017684, 9.257173661936343, 9.573278338766924, 10.34321895800460, 10.83137592976229, 11.43336368092681, 12.46083751452920, 12.62853796793232, 13.08469015358571, 13.86240519599327, 14.39503683217845, 14.88461647192364, 15.31800297857919, 16.05030948641354
