| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 3·15-s + 16-s − 17-s + 18-s + 3·19-s + 3·20-s + 2·22-s − 5·23-s + 24-s + 4·25-s + 27-s + 6·29-s + 3·30-s + 32-s + 2·33-s − 34-s + 36-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 + 0.603·11-s + 0.288·12-s + 0.774·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.688·19-s + 0.670·20-s + 0.426·22-s − 1.04·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s + 1.11·29-s + 0.547·30-s + 0.176·32-s + 0.348·33-s − 0.171·34-s + 1/6·36-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
| \(L(1)\) |
\(\approx\) |
\(5.495337864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.495337864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.158978087243480319183013267571, −7.49585543863322214929777087525, −6.45566131189331170172289197310, −6.21640199086290256231632506715, −5.30626091693421877227863193649, −4.56777762471899702166587452126, −3.70649949821816241940939827033, −2.81460590598354148744553776528, −2.08500818441762413954471190760, −1.25186530448668064429351565160,
1.25186530448668064429351565160, 2.08500818441762413954471190760, 2.81460590598354148744553776528, 3.70649949821816241940939827033, 4.56777762471899702166587452126, 5.30626091693421877227863193649, 6.21640199086290256231632506715, 6.45566131189331170172289197310, 7.49585543863322214929777087525, 8.158978087243480319183013267571
