| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s − 2·9-s + 10-s + 11-s − 12-s − 2·13-s + 15-s + 16-s + 2·18-s + 3·19-s − 20-s − 22-s − 4·23-s + 24-s − 4·25-s + 2·26-s + 5·27-s + 8·29-s − 30-s + 10·31-s − 32-s − 33-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 − 2/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.471·18-s + 0.688·19-s − 0.223·20-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.392·26-s + 0.962·27-s + 1.48·29-s − 0.182·30-s + 1.79·31-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.100902678528243546456276465350, −7.49980657494427080462399167739, −6.55575267412877307711830661234, −6.07746459300176559063050470674, −5.14932522436778583575689845642, −4.35492561871153026724163150401, −3.24283408867439695303836753170, −2.42599213687054455495825999144, −1.09377674289733282160198380336, 0,
1.09377674289733282160198380336, 2.42599213687054455495825999144, 3.24283408867439695303836753170, 4.35492561871153026724163150401, 5.14932522436778583575689845642, 6.07746459300176559063050470674, 6.55575267412877307711830661234, 7.49980657494427080462399167739, 8.100902678528243546456276465350
