| L(s) = 1 | − 2.23·2-s + 3-s + 3.00·4-s − 2.23·6-s − 2.23·8-s + 9-s + 3.00·12-s + 4.47·13-s − 0.999·16-s + 4.47·17-s − 2.23·18-s − 4·23-s − 2.23·24-s − 10.0·26-s + 27-s + 8.94·29-s + 6.70·32-s − 10.0·34-s + 3.00·36-s − 8·37-s + 4.47·39-s − 8.94·41-s − 8.94·43-s + 8.94·46-s − 12·47-s − 0.999·48-s − 7·49-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s − 0.912·6-s − 0.790·8-s + 0.333·9-s + 0.866·12-s + 1.24·13-s − 0.249·16-s + 1.08·17-s − 0.527·18-s − 0.834·23-s − 0.456·24-s − 1.96·26-s + 0.192·27-s + 1.66·29-s + 1.18·32-s − 1.71·34-s + 0.500·36-s − 1.31·37-s + 0.716·39-s − 1.39·41-s − 1.36·43-s + 1.31·46-s − 1.75·47-s − 0.144·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.80278173322157830797801385462, −6.70888002838016970656254523077, −6.57248491214522218758261382077, −5.44617139214084470328659213103, −4.54611268597233469243734731239, −3.50017786910678437681853352710, −2.95685868549119713049748979334, −1.69794876246781604892432061821, −1.34160496575824051053339498352, 0,
1.34160496575824051053339498352, 1.69794876246781604892432061821, 2.95685868549119713049748979334, 3.50017786910678437681853352710, 4.54611268597233469243734731239, 5.44617139214084470328659213103, 6.57248491214522218758261382077, 6.70888002838016970656254523077, 7.80278173322157830797801385462
