| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 2·13-s − 4·14-s + 16-s + 6·17-s − 4·19-s + 2·26-s + 4·28-s + 6·29-s + 8·31-s − 32-s − 6·34-s − 2·37-s + 4·38-s + 6·41-s + 4·43-s + 9·49-s − 2·52-s − 6·53-s − 4·56-s − 6·58-s − 10·61-s − 8·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.392·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.277·52-s − 0.824·53-s − 0.534·56-s − 0.787·58-s − 1.28·61-s − 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196382228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196382228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.91744252348428399819557462541, −10.27409415171032704331958668230, −9.266851904332541042335427380588, −8.128521468164053694071078641328, −7.84831740970949093070141074865, −6.58767052470959898247985953468, −5.37349128799592068356886197367, −4.36655234059998826349058539516, −2.65454187476862090428654397793, −1.28283795797463795719739258116,
1.28283795797463795719739258116, 2.65454187476862090428654397793, 4.36655234059998826349058539516, 5.37349128799592068356886197367, 6.58767052470959898247985953468, 7.84831740970949093070141074865, 8.128521468164053694071078641328, 9.266851904332541042335427380588, 10.27409415171032704331958668230, 10.91744252348428399819557462541
