| L(s) = 1 | − 3-s + 4-s + 7-s + 9-s − 12-s + 16-s − 21-s + 25-s − 27-s + 28-s + 36-s − 2·37-s − 48-s + 49-s + 63-s + 64-s − 75-s − 2·79-s + 81-s − 84-s + 2·97-s + 100-s − 108-s + 2·111-s + 112-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 7-s + 9-s − 12-s + 16-s − 21-s + 25-s − 27-s + 28-s + 36-s − 2·37-s − 48-s + 49-s + 63-s + 64-s − 75-s − 2·79-s + 81-s − 84-s + 2·97-s + 100-s − 108-s + 2·111-s + 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.318939808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318939808\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−8.970035365454316184859119886039, −8.132164708396610233804514713137, −7.26965357009233319984836059413, −6.84051775572323452688870002974, −5.91833266980848018855755014639, −5.25612134274625659774203203883, −4.50656408974110709396398233880, −3.37141723077882067454225868014, −2.10574891386774616326482004588, −1.24221375794159463688402334047,
1.24221375794159463688402334047, 2.10574891386774616326482004588, 3.37141723077882067454225868014, 4.50656408974110709396398233880, 5.25612134274625659774203203883, 5.91833266980848018855755014639, 6.84051775572323452688870002974, 7.26965357009233319984836059413, 8.132164708396610233804514713137, 8.970035365454316184859119886039
