| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 3·11-s + 5·13-s + 2·15-s − 17-s + 6·19-s − 21-s + 4·23-s − 25-s − 27-s − 29-s − 3·33-s − 2·35-s + 8·37-s − 5·39-s − 10·41-s + 4·43-s − 2·45-s + 7·47-s − 6·49-s + 51-s − 2·53-s − 6·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.38·13-s + 0.516·15-s − 0.242·17-s + 1.37·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.185·29-s − 0.522·33-s − 0.338·35-s + 1.31·37-s − 0.800·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.02·47-s − 6/7·49-s + 0.140·51-s − 0.274·53-s − 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.100122370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100122370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−11.43273667210714225198582211926, −10.97068314281864484181970531247, −9.637871756642183477262657920063, −8.654893908187538893836500050806, −7.67887968324846853631830302948, −6.70747891758788855958534164846, −5.63832038652767849241596678261, −4.39775276556313161721819607185, −3.44164669981770176306332772865, −1.19241476492855777568757791158,
1.19241476492855777568757791158, 3.44164669981770176306332772865, 4.39775276556313161721819607185, 5.63832038652767849241596678261, 6.70747891758788855958534164846, 7.67887968324846853631830302948, 8.654893908187538893836500050806, 9.637871756642183477262657920063, 10.97068314281864484181970531247, 11.43273667210714225198582211926
