| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 4·11-s − 2·12-s − 14-s + 16-s − 6·17-s + 18-s − 6·19-s + 2·21-s + 4·22-s + 23-s − 2·24-s + 4·27-s − 28-s + 10·29-s + 4·31-s + 32-s − 8·33-s − 6·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.37·19-s + 0.436·21-s + 0.852·22-s + 0.208·23-s − 0.408·24-s + 0.769·27-s − 0.188·28-s + 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.39·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−6.88735318942094173624207569262, −6.53971592835965782705844673571, −6.33174311279454034272172298867, −5.39506292617571067794559020987, −4.52949482071829905188801506663, −4.29914013210746512672387954271, −3.19585646093781940426000197576, −2.30390877755402852348884298610, −1.20509634533530573283088519986, 0,
1.20509634533530573283088519986, 2.30390877755402852348884298610, 3.19585646093781940426000197576, 4.29914013210746512672387954271, 4.52949482071829905188801506663, 5.39506292617571067794559020987, 6.33174311279454034272172298867, 6.53971592835965782705844673571, 6.88735318942094173624207569262
