| L(s) = 1 | − 5-s + 4·7-s − 6·11-s − 4·13-s − 3·17-s + 7·19-s + 9·23-s + 25-s + 7·31-s − 4·35-s + 2·37-s + 6·41-s − 2·43-s + 9·49-s + 9·53-s + 6·55-s + 12·59-s − 7·61-s + 4·65-s − 2·67-s − 6·71-s + 2·73-s − 24·77-s + 79-s − 9·83-s + 3·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.80·11-s − 1.10·13-s − 0.727·17-s + 1.60·19-s + 1.87·23-s + 1/5·25-s + 1.25·31-s − 0.676·35-s + 0.328·37-s + 0.937·41-s − 0.304·43-s + 9/7·49-s + 1.23·53-s + 0.809·55-s + 1.56·59-s − 0.896·61-s + 0.496·65-s − 0.244·67-s − 0.712·71-s + 0.234·73-s − 2.73·77-s + 0.112·79-s − 0.987·83-s + 0.325·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700553425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700553425\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.906429353426188429522870318186, −8.192207746583714863523309085769, −7.48994188453136730683659056906, −7.16773505985843393013962155051, −5.60918665203988893042610744568, −4.93221836093927602314931363120, −4.58667466503297356148866327247, −3.02936661322764799346381980481, −2.33312000481442186435405366920, −0.862830882447946380721061484065,
0.862830882447946380721061484065, 2.33312000481442186435405366920, 3.02936661322764799346381980481, 4.58667466503297356148866327247, 4.93221836093927602314931363120, 5.60918665203988893042610744568, 7.16773505985843393013962155051, 7.48994188453136730683659056906, 8.192207746583714863523309085769, 8.906429353426188429522870318186
