| L(s) = 1 | − 0.119·3-s − 2.15·5-s + 7-s − 2.98·9-s + 11-s − 13-s + 0.257·15-s − 0.163·17-s − 3.89·19-s − 0.119·21-s + 6.01·23-s − 0.371·25-s + 0.715·27-s + 2.38·29-s + 1.09·31-s − 0.119·33-s − 2.15·35-s + 8.82·37-s + 0.119·39-s + 1.81·41-s + 2.23·43-s + 6.42·45-s − 3.87·47-s + 49-s + 0.0195·51-s + 9.71·53-s − 2.15·55-s + ⋯ |
| L(s) = 1 | − 0.0690·3-s − 0.962·5-s + 0.377·7-s − 0.995·9-s + 0.301·11-s − 0.277·13-s + 0.0664·15-s − 0.0396·17-s − 0.893·19-s − 0.0260·21-s + 1.25·23-s − 0.0742·25-s + 0.137·27-s + 0.442·29-s + 0.196·31-s − 0.0208·33-s − 0.363·35-s + 1.45·37-s + 0.0191·39-s + 0.283·41-s + 0.341·43-s + 0.957·45-s − 0.565·47-s + 0.142·49-s + 0.00273·51-s + 1.33·53-s − 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.119T + 3T^{2} \) |
| 5 | \( 1 + 2.15T + 5T^{2} \) |
| 17 | \( 1 + 0.163T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 - 8.82T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 9.71T + 53T^{2} \) |
| 59 | \( 1 + 9.50T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 + 9.80T + 73T^{2} \) |
| 79 | \( 1 + 9.91T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 3.55T + 97T^{2} \) |
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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
−7.52132320068596989965173458106, −6.88828221644052174915681573071, −6.09664927713992971581599150116, −5.39153256940434523587228298846, −4.53508622877840280454660452840, −4.03655603367192568370188531163, −3.05296306686259236472760729233, −2.40749121242903071054626422604, −1.08971616204818068547373109392, 0,
1.08971616204818068547373109392, 2.40749121242903071054626422604, 3.05296306686259236472760729233, 4.03655603367192568370188531163, 4.53508622877840280454660452840, 5.39153256940434523587228298846, 6.09664927713992971581599150116, 6.88828221644052174915681573071, 7.52132320068596989965173458106
