L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 10-s − 11-s − 12-s − 13-s + 15-s + 16-s + 2·19-s − 20-s − 22-s − 24-s − 26-s + 27-s + 29-s + 30-s − 31-s + 32-s + 33-s + 2·38-s + 39-s − 40-s − 43-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 10-s − 11-s − 12-s − 13-s + 15-s + 16-s + 2·19-s − 20-s − 22-s − 24-s − 26-s + 27-s + 29-s + 30-s − 31-s + 32-s + 33-s + 2·38-s + 39-s − 40-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6792266740\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6792266740\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−13.74254106518691585642797536546, −12.49738155406116875572269725213, −11.84093943455898711558667245967, −11.16700118547112250006640331579, −10.04581052862782411520590533496, −7.927650309728446968100608044272, −7.05540575330027098655761742099, −5.56304821134120925869298444888, −4.78764434142051405584396595400, −3.11568281765995191104117732931,
3.11568281765995191104117732931, 4.78764434142051405584396595400, 5.56304821134120925869298444888, 7.05540575330027098655761742099, 7.927650309728446968100608044272, 10.04581052862782411520590533496, 11.16700118547112250006640331579, 11.84093943455898711558667245967, 12.49738155406116875572269725213, 13.74254106518691585642797536546