| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 11-s − 13-s − 14-s − 15-s − 16-s − 18-s − 19-s + 21-s − 22-s + 24-s + 26-s + 27-s − 29-s + 30-s + 33-s − 35-s − 37-s + 38-s − 39-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 11-s − 13-s − 14-s − 15-s − 16-s − 18-s − 19-s + 21-s − 22-s + 24-s + 26-s + 27-s − 29-s + 30-s + 33-s − 35-s − 37-s + 38-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5463888160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5463888160\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 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
−12.29034442407078492062218497394, −11.32626141128684342298313487562, −10.27543745623849263688080010589, −9.250662911717608031524448689739, −8.511570339456576439539304105406, −7.78841790295682962863972826426, −7.05736954382049469435335613157, −4.71646995003231050683413418513, −3.86731545897061647676023954686, −1.84481602371897264273698743490,
1.84481602371897264273698743490, 3.86731545897061647676023954686, 4.71646995003231050683413418513, 7.05736954382049469435335613157, 7.78841790295682962863972826426, 8.511570339456576439539304105406, 9.250662911717608031524448689739, 10.27543745623849263688080010589, 11.32626141128684342298313487562, 12.29034442407078492062218497394
