| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 2·5-s + 2·6-s + 9-s − 4·10-s − 2·11-s + 2·12-s + 13-s − 2·15-s − 4·16-s + 2·18-s + 19-s − 4·20-s − 4·22-s − 25-s + 2·26-s + 27-s + 4·29-s − 4·30-s + 9·31-s − 8·32-s − 2·33-s + 2·36-s + 3·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s + 1/3·9-s − 1.26·10-s − 0.603·11-s + 0.577·12-s + 0.277·13-s − 0.516·15-s − 16-s + 0.471·18-s + 0.229·19-s − 0.894·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.742·29-s − 0.730·30-s + 1.61·31-s − 1.41·32-s − 0.348·33-s + 1/3·36-s + 0.493·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.134033734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134033734\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27367561777591681476573349415, −12.20598219965913503158804320546, −11.52656811430915887505312945599, −10.18473039565837486570090180668, −8.693773060427938478692796817657, −7.67583845212820398533569804897, −6.39199342358304616891806651452, −4.96455995261269316221263952539, −3.92756051958450667801160124862, −2.81502084362613660119824052187,
2.81502084362613660119824052187, 3.92756051958450667801160124862, 4.96455995261269316221263952539, 6.39199342358304616891806651452, 7.67583845212820398533569804897, 8.693773060427938478692796817657, 10.18473039565837486570090180668, 11.52656811430915887505312945599, 12.20598219965913503158804320546, 13.27367561777591681476573349415
