| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 12·12-s − 26·13-s − 14·14-s + 16·16-s − 18·17-s + 18·18-s + 92·19-s + 21·21-s − 24·24-s − 52·26-s − 27·27-s − 28·28-s − 6·29-s − 4·31-s + 32·32-s − 36·34-s + 36·36-s − 410·37-s + 184·38-s + 78·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.256·17-s + 0.235·18-s + 1.11·19-s + 0.218·21-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.0384·29-s − 0.0231·31-s + 0.176·32-s − 0.181·34-s + 1/6·36-s − 1.82·37-s + 0.785·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 T + p^{3} T^{2} \) |
| 37 | \( 1 + 410 T + p^{3} T^{2} \) |
| 41 | \( 1 - 174 T + p^{3} T^{2} \) |
| 43 | \( 1 + 248 T + p^{3} T^{2} \) |
| 47 | \( 1 + 420 T + p^{3} T^{2} \) |
| 53 | \( 1 + 102 T + p^{3} T^{2} \) |
| 59 | \( 1 + 588 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 + 152 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 610 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1048 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 + 834 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−9.322164137760716687831478872748, −8.115673196878916861722520230065, −7.14543436096929947064088016424, −6.54712717515550900817614991667, −5.50968892295666891307520399970, −4.91809311806139969391244678477, −3.81305438064128803835300895847, −2.84374128880192915325207783499, −1.51158524687156624325709863167, 0,
1.51158524687156624325709863167, 2.84374128880192915325207783499, 3.81305438064128803835300895847, 4.91809311806139969391244678477, 5.50968892295666891307520399970, 6.54712717515550900817614991667, 7.14543436096929947064088016424, 8.115673196878916861722520230065, 9.322164137760716687831478872748
