| L(s) = 1 | − 2·4-s − 5·13-s + 4·16-s + 7·19-s − 5·25-s + 4·31-s + 11·37-s + 8·43-s + 10·52-s + 61-s − 8·64-s + 5·67-s + 7·73-s − 14·76-s + 17·79-s + 19·97-s + 10·100-s + 13·103-s + 2·109-s + ⋯ |
| L(s) = 1 | − 4-s − 1.38·13-s + 16-s + 1.60·19-s − 25-s + 0.718·31-s + 1.80·37-s + 1.21·43-s + 1.38·52-s + 0.128·61-s − 64-s + 0.610·67-s + 0.819·73-s − 1.60·76-s + 1.91·79-s + 1.92·97-s + 100-s + 1.28·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156536542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156536542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−9.692549886033731115950097335871, −9.003116301093652454576632877528, −7.83187954508823562149152602221, −7.53309190588908388391290020645, −6.19032093295709689179707806358, −5.27555176123571387068189863753, −4.61317128196167712173450637604, −3.61858250543880863713873154871, −2.48005247853672969845362220771, −0.794908582887598964605828839390,
0.794908582887598964605828839390, 2.48005247853672969845362220771, 3.61858250543880863713873154871, 4.61317128196167712173450637604, 5.27555176123571387068189863753, 6.19032093295709689179707806358, 7.53309190588908388391290020645, 7.83187954508823562149152602221, 9.003116301093652454576632877528, 9.692549886033731115950097335871
