| L(s) = 1 | + 2·2-s − 4·4-s − 24·8-s + 10·11-s + 80·13-s − 16·16-s − 7·17-s − 113·19-s + 20·22-s + 81·23-s + 160·26-s − 220·29-s − 189·31-s + 160·32-s − 14·34-s − 170·37-s − 226·38-s − 130·41-s − 10·43-s − 40·44-s + 162·46-s − 160·47-s − 343·49-s − 320·52-s − 631·53-s − 440·58-s − 560·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.274·11-s + 1.70·13-s − 1/4·16-s − 0.0998·17-s − 1.36·19-s + 0.193·22-s + 0.734·23-s + 1.20·26-s − 1.40·29-s − 1.09·31-s + 0.883·32-s − 0.0706·34-s − 0.755·37-s − 0.964·38-s − 0.495·41-s − 0.0354·43-s − 0.137·44-s + 0.519·46-s − 0.496·47-s − 49-s − 0.853·52-s − 1.63·53-s − 0.996·58-s − 1.23·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 10 T + p^{3} T^{2} \) |
| 13 | \( 1 - 80 T + p^{3} T^{2} \) |
| 17 | \( 1 + 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 113 T + p^{3} T^{2} \) |
| 23 | \( 1 - 81 T + p^{3} T^{2} \) |
| 29 | \( 1 + 220 T + p^{3} T^{2} \) |
| 31 | \( 1 + 189 T + p^{3} T^{2} \) |
| 37 | \( 1 + 170 T + p^{3} T^{2} \) |
| 41 | \( 1 + 130 T + p^{3} T^{2} \) |
| 43 | \( 1 + 10 T + p^{3} T^{2} \) |
| 47 | \( 1 + 160 T + p^{3} T^{2} \) |
| 53 | \( 1 + 631 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 229 T + p^{3} T^{2} \) |
| 67 | \( 1 + 750 T + p^{3} T^{2} \) |
| 71 | \( 1 - 890 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 + 27 T + p^{3} T^{2} \) |
| 83 | \( 1 + 429 T + p^{3} T^{2} \) |
| 89 | \( 1 + 750 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1480 T + p^{3} T^{2} \) |
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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
−9.473350142016090354261734034259, −8.847916902988192892868866492728, −8.096381650860031042439033533846, −6.68555216320812904796588143701, −5.98558334599506096651554746767, −5.02079544306787479412824616499, −3.97383276555307234043111630754, −3.31737701567894507880431541512, −1.63729010067989467341178877996, 0,
1.63729010067989467341178877996, 3.31737701567894507880431541512, 3.97383276555307234043111630754, 5.02079544306787479412824616499, 5.98558334599506096651554746767, 6.68555216320812904796588143701, 8.096381650860031042439033533846, 8.847916902988192892868866492728, 9.473350142016090354261734034259
