| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 3·11-s + 12-s + 3·13-s − 15-s + 16-s − 17-s − 18-s − 6·19-s − 20-s − 3·22-s − 2·23-s − 24-s − 4·25-s − 3·26-s + 27-s + 6·29-s + 30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.832·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.639·22-s − 0.417·23-s − 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.343061139815913949864945799357, −7.16706591064391616236631338393, −6.66923523237448997238754062583, −5.99026484408832790522330358390, −4.81273196701121948518979250800, −3.86269188277869207036272840126, −3.41572377468383889013834882480, −2.15365618234657513176883299181, −1.44860584742228176636039930443, 0,
1.44860584742228176636039930443, 2.15365618234657513176883299181, 3.41572377468383889013834882480, 3.86269188277869207036272840126, 4.81273196701121948518979250800, 5.99026484408832790522330358390, 6.66923523237448997238754062583, 7.16706591064391616236631338393, 8.343061139815913949864945799357
