| L(s) = 1 | + 5-s + 7-s + 9-s − 11-s − 17-s + 19-s − 2·23-s + 35-s − 43-s + 45-s + 47-s − 55-s + 61-s + 63-s − 73-s − 77-s + 81-s + 2·83-s − 85-s + 95-s − 99-s − 2·101-s − 2·115-s − 119-s + ⋯ |
| L(s) = 1 | + 5-s + 7-s + 9-s − 11-s − 17-s + 19-s − 2·23-s + 35-s − 43-s + 45-s + 47-s − 55-s + 61-s + 63-s − 73-s − 77-s + 81-s + 2·83-s − 85-s + 95-s − 99-s − 2·101-s − 2·115-s − 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.356183536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356183536\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )^{2} \) |
| 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
−10.03427034591441808879612149544, −9.255098424372228179752418993373, −8.172405046070473170772047601897, −7.59999010431101647124163945265, −6.58501746998425739761414145405, −5.62273489322617195952942870189, −4.91269600717961515168772884863, −3.96972632932219871467433158078, −2.38790724946517221009219670095, −1.63850364422660229804002430304,
1.63850364422660229804002430304, 2.38790724946517221009219670095, 3.96972632932219871467433158078, 4.91269600717961515168772884863, 5.62273489322617195952942870189, 6.58501746998425739761414145405, 7.59999010431101647124163945265, 8.172405046070473170772047601897, 9.255098424372228179752418993373, 10.03427034591441808879612149544
