| L(s) = 1 | − 4-s − 5-s + 2·11-s + 16-s + 19-s + 20-s + 25-s − 2·44-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s − 80-s − 95-s − 100-s + 2·101-s + ⋯ |
| L(s) = 1 | − 4-s − 5-s + 2·11-s + 16-s + 19-s + 20-s + 25-s − 2·44-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s − 80-s − 95-s − 100-s + 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7508262571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7508262571\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.28779421471673768524189790270, −9.260276857120472535061526031545, −8.902620918268777374754762733311, −7.900857310226984383949826537656, −7.07786177591191312722904428978, −6.02587532039734194154641480246, −4.81742521785523930316284244708, −4.02943525995896816813128527339, −3.33358023614207416943421030004, −1.14991567077063894137476213631,
1.14991567077063894137476213631, 3.33358023614207416943421030004, 4.02943525995896816813128527339, 4.81742521785523930316284244708, 6.02587532039734194154641480246, 7.07786177591191312722904428978, 7.900857310226984383949826537656, 8.902620918268777374754762733311, 9.260276857120472535061526031545, 10.28779421471673768524189790270
