| L(s) = 1 | − 0.760·2-s + 3-s − 1.42·4-s + 1.94·5-s − 0.760·6-s − 5.18·7-s + 2.60·8-s + 9-s − 1.47·10-s + 11-s − 1.42·12-s − 4.23·13-s + 3.94·14-s + 1.94·15-s + 0.861·16-s − 2.28·17-s − 0.760·18-s − 19-s − 2.76·20-s − 5.18·21-s − 0.760·22-s − 5·23-s + 2.60·24-s − 1.22·25-s + 3.22·26-s + 27-s + 7.36·28-s + ⋯ |
| L(s) = 1 | − 0.538·2-s + 0.577·3-s − 0.710·4-s + 0.868·5-s − 0.310·6-s − 1.95·7-s + 0.920·8-s + 0.333·9-s − 0.467·10-s + 0.301·11-s − 0.410·12-s − 1.17·13-s + 1.05·14-s + 0.501·15-s + 0.215·16-s − 0.553·17-s − 0.179·18-s − 0.229·19-s − 0.617·20-s − 1.13·21-s − 0.162·22-s − 1.04·23-s + 0.531·24-s − 0.245·25-s + 0.632·26-s + 0.192·27-s + 1.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.760T + 2T^{2} \) |
| 5 | \( 1 - 1.94T + 5T^{2} \) |
| 7 | \( 1 + 5.18T + 7T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + 9.72T + 37T^{2} \) |
| 41 | \( 1 + 0.239T + 41T^{2} \) |
| 43 | \( 1 + 0.578T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 - 7.03T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 6.66T + 89T^{2} \) |
| 97 | \( 1 + 8.74T + 97T^{2} \) |
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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
−9.945960558921735708211469519964, −9.380918201746149763267946659055, −8.797656632967591469260696952254, −7.58100862009213872539159022626, −6.66591394883088851365464324913, −5.75014661566536491598819371277, −4.40703345469512683488161062797, −3.31743313767316729267868515798, −2.07609450278487697987943329898, 0,
2.07609450278487697987943329898, 3.31743313767316729267868515798, 4.40703345469512683488161062797, 5.75014661566536491598819371277, 6.66591394883088851365464324913, 7.58100862009213872539159022626, 8.797656632967591469260696952254, 9.380918201746149763267946659055, 9.945960558921735708211469519964
