| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s − 10-s + 12-s + 15-s + 16-s + 20-s + 23-s − 24-s + 25-s − 27-s − 29-s − 30-s − 32-s − 40-s − 41-s + 43-s − 46-s − 2·47-s + 48-s − 50-s + 54-s + 58-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s − 10-s + 12-s + 15-s + 16-s + 20-s + 23-s − 24-s + 25-s − 27-s − 29-s − 30-s − 32-s − 40-s − 41-s + 43-s − 46-s − 2·47-s + 48-s − 50-s + 54-s + 58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.041211610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041211610\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.869693084453220128949828844026, −9.307040558155344001394426457467, −8.719440002860592990547946555940, −7.930338811691618361306348548308, −7.03237660848538221891905703905, −6.14748564902986047144463161580, −5.17737013649407497594184163409, −3.44066727284704930723340523309, −2.57802748041299452045427181021, −1.61736189099351821913666552954,
1.61736189099351821913666552954, 2.57802748041299452045427181021, 3.44066727284704930723340523309, 5.17737013649407497594184163409, 6.14748564902986047144463161580, 7.03237660848538221891905703905, 7.930338811691618361306348548308, 8.719440002860592990547946555940, 9.307040558155344001394426457467, 9.869693084453220128949828844026
