| L(s) = 1 | − 2·2-s + 4·4-s + 16·7-s − 8·8-s − 12·11-s − 38·13-s − 32·14-s + 16·16-s − 126·17-s + 20·19-s + 24·22-s + 168·23-s + 76·26-s + 64·28-s − 30·29-s − 88·31-s − 32·32-s + 252·34-s − 254·37-s − 40·38-s − 42·41-s + 52·43-s − 48·44-s − 336·46-s − 96·47-s − 87·49-s − 152·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.863·7-s − 0.353·8-s − 0.328·11-s − 0.810·13-s − 0.610·14-s + 1/4·16-s − 1.79·17-s + 0.241·19-s + 0.232·22-s + 1.52·23-s + 0.573·26-s + 0.431·28-s − 0.192·29-s − 0.509·31-s − 0.176·32-s + 1.27·34-s − 1.12·37-s − 0.170·38-s − 0.159·41-s + 0.184·43-s − 0.164·44-s − 1.07·46-s − 0.297·47-s − 0.253·49-s − 0.405·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 88 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 538 T + p^{3} T^{2} \) |
| 67 | \( 1 + 884 T + p^{3} T^{2} \) |
| 71 | \( 1 + 792 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1154 T + p^{3} T^{2} \) |
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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
−10.28980993856618728252432495605, −9.142352126398637491680774504765, −8.572644456961224873468537629519, −7.46232944013583376836818811775, −6.81983780086633644841981277146, −5.39581449739666887816713537592, −4.47421463394464483428606675813, −2.79284878392979081847905451390, −1.64382833567453141638750952424, 0,
1.64382833567453141638750952424, 2.79284878392979081847905451390, 4.47421463394464483428606675813, 5.39581449739666887816713537592, 6.81983780086633644841981277146, 7.46232944013583376836818811775, 8.572644456961224873468537629519, 9.142352126398637491680774504765, 10.28980993856618728252432495605
