L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s + 7-s − 8-s + 9-s − 2·10-s + 11-s + 12-s − 13-s − 14-s + 2·15-s + 16-s − 17-s − 18-s − 4·19-s + 2·20-s + 21-s − 22-s − 6·23-s − 24-s − 25-s + 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.45513863218395, −15.86542920284487, −15.17669314055710, −14.76746638734714, −14.19978551636194, −13.54183588003769, −13.25158442036799, −12.28380043768303, −11.94563405264425, −11.13764057589382, −10.46883989397217, −10.06347690871370, −9.462299488794444, −8.947687459126287, −8.373047191561505, −7.805227960114998, −7.148183830551460, −6.420137235206918, −5.912845580357937, −5.136636276652032, −4.236400694474854, −3.588574780045693, −2.469132827969466, −2.059695776447628, −1.361902187761392, 0,
1.361902187761392, 2.059695776447628, 2.469132827969466, 3.588574780045693, 4.236400694474854, 5.136636276652032, 5.912845580357937, 6.420137235206918, 7.148183830551460, 7.805227960114998, 8.373047191561505, 8.947687459126287, 9.462299488794444, 10.06347690871370, 10.46883989397217, 11.13764057589382, 11.94563405264425, 12.28380043768303, 13.25158442036799, 13.54183588003769, 14.19978551636194, 14.76746638734714, 15.17669314055710, 15.86542920284487, 16.45513863218395