| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s + 12-s + 13-s − 14-s − 2·15-s + 16-s − 3·17-s + 18-s + 19-s − 2·20-s − 21-s − 23-s + 24-s − 25-s + 26-s + 27-s − 28-s − 3·29-s − 2·30-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 + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.218·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.95763284521463, −13.30891699197946, −12.90279067659171, −12.49786923713496, −11.99152586226205, −11.43100575328430, −11.04102792822999, −10.63035984846686, −9.848044529776768, −9.452842038665142, −8.892625949156273, −8.325911693670763, −7.810454196968385, −7.408958915908068, −6.893494151265016, −6.325466004675683, −5.755605496130261, −5.202341829944994, −4.412408128691559, −3.969681566330218, −3.761666927557368, −2.888060976208240, −2.534030261217084, −1.764397924381503, −0.9198066984003936, 0,
0.9198066984003936, 1.764397924381503, 2.534030261217084, 2.888060976208240, 3.761666927557368, 3.969681566330218, 4.412408128691559, 5.202341829944994, 5.755605496130261, 6.325466004675683, 6.893494151265016, 7.408958915908068, 7.810454196968385, 8.325911693670763, 8.892625949156273, 9.452842038665142, 9.848044529776768, 10.63035984846686, 11.04102792822999, 11.43100575328430, 11.99152586226205, 12.49786923713496, 12.90279067659171, 13.30891699197946, 13.95763284521463
