| L(s) = 1 | − 5-s + 6·11-s + 2·13-s + 2·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 8·37-s + 2·41-s + 43-s + 12·47-s − 7·49-s − 8·53-s − 6·55-s + 2·59-s − 6·61-s − 2·65-s + 4·67-s + 12·71-s + 12·83-s − 6·89-s − 2·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.80·11-s + 0.554·13-s + 0.458·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 1.31·37-s + 0.312·41-s + 0.152·43-s + 1.75·47-s − 49-s − 1.09·53-s − 0.809·55-s + 0.260·59-s − 0.768·61-s − 0.248·65-s + 0.488·67-s + 1.42·71-s + 1.31·83-s − 0.635·89-s − 0.205·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224762174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224762174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.12371681043377, −14.44287503422662, −14.09874271514048, −13.73291823491869, −12.83669474131616, −12.36658663201698, −11.97656842012117, −11.24728862480851, −11.09229214624759, −10.26960199019503, −9.607024827505790, −9.048071764977980, −8.776816020223369, −7.940961606549164, −7.431980279559366, −6.800882089569132, −6.279186258912906, −5.691549378730393, −4.955333062657604, −4.138140205081537, −3.711067846627781, −3.262935084338896, −2.058045794368415, −1.490081474963939, −0.5922272181753785,
0.5922272181753785, 1.490081474963939, 2.058045794368415, 3.262935084338896, 3.711067846627781, 4.138140205081537, 4.955333062657604, 5.691549378730393, 6.279186258912906, 6.800882089569132, 7.431980279559366, 7.940961606549164, 8.776816020223369, 9.048071764977980, 9.607024827505790, 10.26960199019503, 11.09229214624759, 11.24728862480851, 11.97656842012117, 12.36658663201698, 12.83669474131616, 13.73291823491869, 14.09874271514048, 14.44287503422662, 15.12371681043377
