L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 2·7-s + 2·10-s − 3·11-s − 5·13-s + 4·14-s − 4·16-s − 17-s − 19-s − 2·20-s + 6·22-s − 7·23-s − 4·25-s + 10·26-s − 4·28-s + 6·29-s + 4·31-s + 8·32-s + 2·34-s + 2·35-s + 10·37-s + 2·38-s + 9·41-s + 43-s − 6·44-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s + 0.632·10-s − 0.904·11-s − 1.38·13-s + 1.06·14-s − 16-s − 0.242·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 1.45·23-s − 4/5·25-s + 1.96·26-s − 0.755·28-s + 1.11·29-s + 0.718·31-s + 1.41·32-s + 0.342·34-s + 0.338·35-s + 1.64·37-s + 0.324·38-s + 1.40·41-s + 0.152·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.29040567291484407710402327749, −11.21201107323355823458653573304, −9.992759799763264145998704328660, −9.673767858002872067860717754239, −8.193497167062057609442371334624, −7.62717436015796243936248846504, −6.35321842859344777216335456735, −4.54009421268158840480926538889, −2.52748238071021851087833645012, 0,
2.52748238071021851087833645012, 4.54009421268158840480926538889, 6.35321842859344777216335456735, 7.62717436015796243936248846504, 8.193497167062057609442371334624, 9.673767858002872067860717754239, 9.992759799763264145998704328660, 11.21201107323355823458653573304, 12.29040567291484407710402327749