| L(s) = 1 | + 5-s + 2·7-s − 4·13-s − 6·17-s + 19-s − 6·23-s + 25-s − 6·29-s + 8·31-s + 2·35-s − 4·37-s + 6·41-s − 10·43-s − 6·47-s − 3·49-s − 12·53-s − 12·59-s − 10·61-s − 4·65-s − 4·67-s + 14·73-s + 8·79-s + 18·83-s − 6·85-s + 6·89-s − 8·91-s + 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.10·13-s − 1.45·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.657·37-s + 0.937·41-s − 1.52·43-s − 0.875·47-s − 3/7·49-s − 1.64·53-s − 1.56·59-s − 1.28·61-s − 0.496·65-s − 0.488·67-s + 1.63·73-s + 0.900·79-s + 1.97·83-s − 0.650·85-s + 0.635·89-s − 0.838·91-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.060339332527468493540374098862, −7.68821375990419253757950875881, −6.60989678548919112531977342107, −6.11107864213354438097895517691, −4.82278753305875115493484570808, −4.76844915470858687612104669291, −3.46294620859648001986657984801, −2.32280864169618120360581494828, −1.69690859840940500979154966414, 0,
1.69690859840940500979154966414, 2.32280864169618120360581494828, 3.46294620859648001986657984801, 4.76844915470858687612104669291, 4.82278753305875115493484570808, 6.11107864213354438097895517691, 6.60989678548919112531977342107, 7.68821375990419253757950875881, 8.060339332527468493540374098862
