| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 14-s + 16-s − 18-s − 19-s + 21-s + 24-s + 25-s − 27-s − 28-s − 32-s + 36-s + 38-s − 42-s − 48-s + 49-s − 50-s + 54-s + 56-s + 57-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 14-s + 16-s − 18-s − 19-s + 21-s + 24-s + 25-s − 27-s − 28-s − 32-s + 36-s + 38-s − 42-s − 48-s + 49-s − 50-s + 54-s + 56-s + 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4368988982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4368988982\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 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
−8.979035706155505803952065430972, −8.150464730634850748141083425002, −7.19075894136685679709975730798, −6.66130894003186671942257507277, −6.11323428179343422463815085730, −5.28501025270357429808017510472, −4.16250651262072087135958707375, −3.12819573497556477790001754767, −2.01855626480808879590384798567, −0.69928849855423380692273540097,
0.69928849855423380692273540097, 2.01855626480808879590384798567, 3.12819573497556477790001754767, 4.16250651262072087135958707375, 5.28501025270357429808017510472, 6.11323428179343422463815085730, 6.66130894003186671942257507277, 7.19075894136685679709975730798, 8.150464730634850748141083425002, 8.979035706155505803952065430972
