| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s − 15-s + 16-s − 18-s − 20-s − 22-s − 24-s + 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 36-s + 40-s + 44-s − 45-s + 48-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s − 15-s + 16-s − 18-s − 20-s − 22-s − 24-s + 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 36-s + 40-s + 44-s − 45-s + 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8878209655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8878209655\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.845009634973321026496675217805, −8.935658996603469370744788991884, −8.466043914714623806449997257810, −7.68388755356423240950892232763, −7.04341004523214801677739723124, −6.17944432688854524720630031163, −4.49523384661317097794370744562, −3.60676993800065472094331009730, −2.66085340046447725060743817189, −1.30039397022937313378303439283,
1.30039397022937313378303439283, 2.66085340046447725060743817189, 3.60676993800065472094331009730, 4.49523384661317097794370744562, 6.17944432688854524720630031163, 7.04341004523214801677739723124, 7.68388755356423240950892232763, 8.466043914714623806449997257810, 8.935658996603469370744788991884, 9.845009634973321026496675217805
