| L(s) = 1 | − 3-s − 7-s + 9-s − 2·13-s + 19-s + 21-s + 4·23-s − 5·25-s − 27-s + 8·31-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s − 2·47-s + 49-s − 12·53-s − 57-s − 12·59-s − 10·61-s − 63-s + 4·67-s − 4·69-s + 10·71-s − 10·73-s + 5·75-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.229·19-s + 0.218·21-s + 0.834·23-s − 25-s − 0.192·27-s + 1.43·31-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s − 1.64·53-s − 0.132·57-s − 1.56·59-s − 1.28·61-s − 0.125·63-s + 0.488·67-s − 0.481·69-s + 1.18·71-s − 1.17·73-s + 0.577·75-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.126485074784209528372605629917, −7.58793424113764205292857508887, −6.59525404429396431021771083039, −6.19114585959665152648965844514, −5.14223700146096769131439504731, −4.59530717109858234145861499155, −3.51436727557375149062532492690, −2.60932181779375543690385011448, −1.35049802400259995052643618295, 0,
1.35049802400259995052643618295, 2.60932181779375543690385011448, 3.51436727557375149062532492690, 4.59530717109858234145861499155, 5.14223700146096769131439504731, 6.19114585959665152648965844514, 6.59525404429396431021771083039, 7.58793424113764205292857508887, 8.126485074784209528372605629917
