| L(s) = 1 | + 0.592·3-s + 0.0231·5-s + 0.0195·7-s − 2.64·9-s + 3.14·11-s − 1.84·13-s + 0.0137·15-s + 7.89·17-s + 0.196·19-s + 0.0115·21-s − 0.498·23-s − 4.99·25-s − 3.34·27-s + 8.94·29-s + 6.43·31-s + 1.86·33-s + 0.000453·35-s − 2.68·37-s − 1.09·39-s + 4.50·41-s − 8.62·43-s − 0.0614·45-s + 5.60·47-s − 6.99·49-s + 4.67·51-s + 0.825·53-s + 0.0730·55-s + ⋯ |
| L(s) = 1 | + 0.341·3-s + 0.0103·5-s + 0.00739·7-s − 0.883·9-s + 0.949·11-s − 0.511·13-s + 0.00354·15-s + 1.91·17-s + 0.0451·19-s + 0.00252·21-s − 0.103·23-s − 0.999·25-s − 0.643·27-s + 1.66·29-s + 1.15·31-s + 0.324·33-s + 7.66e−5·35-s − 0.441·37-s − 0.174·39-s + 0.703·41-s − 1.31·43-s − 0.00915·45-s + 0.816·47-s − 0.999·49-s + 0.654·51-s + 0.113·53-s + 0.00984·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.252849571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.252849571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.592T + 3T^{2} \) |
| 5 | \( 1 - 0.0231T + 5T^{2} \) |
| 7 | \( 1 - 0.0195T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 + 1.84T + 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 - 0.196T + 19T^{2} \) |
| 23 | \( 1 + 0.498T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 + 8.62T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 - 0.825T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 - 1.23T + 71T^{2} \) |
| 73 | \( 1 + 1.99T + 73T^{2} \) |
| 79 | \( 1 - 5.26T + 79T^{2} \) |
| 83 | \( 1 - 7.84T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 9.59T + 97T^{2} \) |
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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
−7.993267408554992774655366712321, −7.62548828511910364612266377598, −6.52705269188264850796541799597, −6.04728203384980851576905240451, −5.21686001640689795753184374999, −4.44471817711766288779397045446, −3.42441635057992677201504807438, −2.97309389280366318243712070772, −1.86504422908824418132964105474, −0.78865089828851549796236314839,
0.78865089828851549796236314839, 1.86504422908824418132964105474, 2.97309389280366318243712070772, 3.42441635057992677201504807438, 4.44471817711766288779397045446, 5.21686001640689795753184374999, 6.04728203384980851576905240451, 6.52705269188264850796541799597, 7.62548828511910364612266377598, 7.993267408554992774655366712321
