| L(s) = 1 | − 0.896i·5-s + 1.26i·7-s − 4.24·11-s + 13-s + 0.896i·17-s + 4.73i·19-s + 7.34·23-s + 4.19·25-s + 8.24i·29-s + 6i·31-s + 1.13·35-s + 1.19·37-s − 7.34i·41-s + 8.19i·43-s − 11.5·47-s + ⋯ |
| L(s) = 1 | − 0.400i·5-s + 0.479i·7-s − 1.27·11-s + 0.277·13-s + 0.217i·17-s + 1.08i·19-s + 1.53·23-s + 0.839·25-s + 1.53i·29-s + 1.07i·31-s + 0.192·35-s + 0.196·37-s − 1.14i·41-s + 1.24i·43-s − 1.69·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.377161937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377161937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.896iT - 5T^{2} \) |
| 7 | \( 1 - 1.26iT - 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.896iT - 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 - 8.24iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 + 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 8.19iT - 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 7.34iT - 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 - 10.7iT - 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 4.73iT - 79T^{2} \) |
| 83 | \( 1 + 8.48T + 83T^{2} \) |
| 89 | \( 1 - 11.8iT - 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.815686153391986363342497380994, −8.751381298006495212647522044672, −8.405676027105561117306961173332, −7.36710814579707623430462450368, −6.51671268754033087486196322485, −5.29420071046552904404342310429, −5.05392307421370389158725890363, −3.58494987320886144053407202266, −2.65975586666681119550042584927, −1.29067306132049335121077859207,
0.62636944169353104747156097358, 2.40706715761136472726768092491, 3.19032840110733127513842602433, 4.47033763115544233288760942394, 5.21300858817396136466823437095, 6.29639240678561329973196592109, 7.13850021459850954556419075921, 7.78582087158382148101353886814, 8.693085768884526761113708662547, 9.597286788047935725657871429710
