L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s − 6·11-s − 12-s − 5·13-s + 3·15-s + 16-s + 17-s − 18-s + 4·19-s − 3·20-s + 6·22-s + 24-s + 4·25-s + 5·26-s − 27-s + 9·29-s − 3·30-s + 31-s − 32-s + 6·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.80·11-s − 0.288·12-s − 1.38·13-s + 0.774·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.670·20-s + 1.27·22-s + 0.204·24-s + 4/5·25-s + 0.980·26-s − 0.192·27-s + 1.67·29-s − 0.547·30-s + 0.179·31-s − 0.176·32-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.76482017045702025069531101302, −7.54245587812423913615992420133, −6.66803978192936400250652851273, −5.67319404871497261989642156121, −4.91880215321054947673162787354, −4.33261234288388326343282393623, −3.03530548823916719414017780997, −2.52182957178913491491072231079, −0.873514584861722047688549187758, 0,
0.873514584861722047688549187758, 2.52182957178913491491072231079, 3.03530548823916719414017780997, 4.33261234288388326343282393623, 4.91880215321054947673162787354, 5.67319404871497261989642156121, 6.66803978192936400250652851273, 7.54245587812423913615992420133, 7.76482017045702025069531101302