| L(s) = 1 | − 2.75·3-s − 3.57·7-s + 4.57·9-s − 11-s + 13-s + 0.751·17-s + 2.50·19-s + 9.82·21-s − 5.75·23-s − 4.32·27-s − 4.07·29-s + 6.14·31-s + 2.75·33-s − 2.81·37-s − 2.75·39-s + 1.18·41-s − 7.68·43-s + 9.82·47-s + 5.75·49-s − 2.06·51-s + 14.2·53-s − 6.88·57-s + 9.82·59-s + 7.07·61-s − 16.3·63-s + 14.6·67-s + 15.8·69-s + ⋯ |
| L(s) = 1 | − 1.58·3-s − 1.34·7-s + 1.52·9-s − 0.301·11-s + 0.277·13-s + 0.182·17-s + 0.574·19-s + 2.14·21-s − 1.19·23-s − 0.831·27-s − 0.756·29-s + 1.10·31-s + 0.478·33-s − 0.463·37-s − 0.440·39-s + 0.184·41-s − 1.17·43-s + 1.43·47-s + 0.821·49-s − 0.289·51-s + 1.95·53-s − 0.912·57-s + 1.27·59-s + 0.905·61-s − 2.05·63-s + 1.78·67-s + 1.90·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.751T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 3.89T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 - 7.42T + 89T^{2} \) |
| 97 | \( 1 + 0.609T + 97T^{2} \) |
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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
−7.83806082026236501285341925938, −6.89960401340917898243691687146, −6.55503138355393978478244201876, −5.63914245047047525276967358897, −5.43895531124835364822869277570, −4.24647931767769711587824952623, −3.55723602442372565435994230888, −2.41439020979404007672474579879, −0.974567228688667989692611856426, 0,
0.974567228688667989692611856426, 2.41439020979404007672474579879, 3.55723602442372565435994230888, 4.24647931767769711587824952623, 5.43895531124835364822869277570, 5.63914245047047525276967358897, 6.55503138355393978478244201876, 6.89960401340917898243691687146, 7.83806082026236501285341925938
