| L(s) = 1 | + 3·3-s − 8·4-s − 7·7-s + 9·9-s + 42·11-s − 24·12-s − 20·13-s + 64·16-s − 66·17-s + 38·19-s − 21·21-s − 12·23-s + 27·27-s + 56·28-s − 258·29-s + 146·31-s + 126·33-s − 72·36-s − 434·37-s − 60·39-s − 282·41-s − 20·43-s − 336·44-s + 72·47-s + 192·48-s + 49·49-s − 198·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 1.15·11-s − 0.577·12-s − 0.426·13-s + 16-s − 0.941·17-s + 0.458·19-s − 0.218·21-s − 0.108·23-s + 0.192·27-s + 0.377·28-s − 1.65·29-s + 0.845·31-s + 0.664·33-s − 1/3·36-s − 1.92·37-s − 0.246·39-s − 1.07·41-s − 0.0709·43-s − 1.15·44-s + 0.223·47-s + 0.577·48-s + 1/7·49-s − 0.543·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 258 T + p^{3} T^{2} \) |
| 31 | \( 1 - 146 T + p^{3} T^{2} \) |
| 37 | \( 1 + 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 336 T + p^{3} T^{2} \) |
| 59 | \( 1 + 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 682 T + p^{3} T^{2} \) |
| 67 | \( 1 + 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 810 T + p^{3} T^{2} \) |
| 73 | \( 1 - 124 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1208 T + p^{3} T^{2} \) |
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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
−9.660480682934644811494424762636, −9.198263605310083470927787310436, −8.446844434691486110679714658395, −7.36657299145429705693745425736, −6.38185322371920146910964906708, −5.10879224059252464092674021945, −4.10762185843126572820557107934, −3.26339512374771722020879159902, −1.63394024708422573519915901457, 0,
1.63394024708422573519915901457, 3.26339512374771722020879159902, 4.10762185843126572820557107934, 5.10879224059252464092674021945, 6.38185322371920146910964906708, 7.36657299145429705693745425736, 8.446844434691486110679714658395, 9.198263605310083470927787310436, 9.660480682934644811494424762636
