| L(s) = 1 | + 3-s − 2·4-s + 9-s + 6·11-s − 2·12-s + 5·13-s + 4·16-s − 6·17-s − 19-s − 23-s − 5·25-s + 27-s + 6·29-s + 5·31-s + 6·33-s − 2·36-s − 7·37-s + 5·39-s − 43-s − 12·44-s + 6·47-s + 4·48-s − 6·51-s − 10·52-s + 12·53-s − 57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 1.38·13-s + 16-s − 1.45·17-s − 0.229·19-s − 0.208·23-s − 25-s + 0.192·27-s + 1.11·29-s + 0.898·31-s + 1.04·33-s − 1/3·36-s − 1.15·37-s + 0.800·39-s − 0.152·43-s − 1.80·44-s + 0.875·47-s + 0.577·48-s − 0.840·51-s − 1.38·52-s + 1.64·53-s − 0.132·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.105694926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105694926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.625037634372769662773359089261, −8.319995239045270419575059284670, −7.03440325934198167051172505911, −6.42986666345209608160897643920, −5.64830121119480539528704598682, −4.27498360461651945204072901678, −4.17644875441450813916070086237, −3.27750005862054764369574055987, −1.90907078639894873995986034777, −0.897757036926060875786439504030,
0.897757036926060875786439504030, 1.90907078639894873995986034777, 3.27750005862054764369574055987, 4.17644875441450813916070086237, 4.27498360461651945204072901678, 5.64830121119480539528704598682, 6.42986666345209608160897643920, 7.03440325934198167051172505911, 8.319995239045270419575059284670, 8.625037634372769662773359089261
