| L(s) = 1 | − 2·3-s + 9-s + 4·11-s − 2·13-s + 2·19-s + 4·23-s + 4·27-s + 10·29-s + 4·31-s − 8·33-s + 2·37-s + 4·39-s − 12·41-s + 4·43-s − 4·47-s − 2·53-s − 4·57-s + 10·59-s + 6·61-s − 4·67-s − 8·69-s − 12·71-s + 4·73-s − 4·79-s − 11·81-s − 14·83-s − 20·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.458·19-s + 0.834·23-s + 0.769·27-s + 1.85·29-s + 0.718·31-s − 1.39·33-s + 0.328·37-s + 0.640·39-s − 1.87·41-s + 0.609·43-s − 0.583·47-s − 0.274·53-s − 0.529·57-s + 1.30·59-s + 0.768·61-s − 0.488·67-s − 0.963·69-s − 1.42·71-s + 0.468·73-s − 0.450·79-s − 1.22·81-s − 1.53·83-s − 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391905900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391905900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.43929637387081210967245648289, −6.75905354590784375569830745212, −6.40726112677636177519562126412, −5.65004317160530344615301939060, −4.88820229370690310115244040584, −4.49503486194408611545297950053, −3.43427166997574220707704307994, −2.65697201280074264644927023751, −1.40229049634602024409274708765, −0.65465875421010450544495538742,
0.65465875421010450544495538742, 1.40229049634602024409274708765, 2.65697201280074264644927023751, 3.43427166997574220707704307994, 4.49503486194408611545297950053, 4.88820229370690310115244040584, 5.65004317160530344615301939060, 6.40726112677636177519562126412, 6.75905354590784375569830745212, 7.43929637387081210967245648289
