| L(s) = 1 | − 2·2-s + 4·4-s + 19·7-s − 8·8-s + 12·11-s − 50·13-s − 38·14-s + 16·16-s + 126·17-s + 29·19-s − 24·22-s − 18·23-s + 100·26-s + 76·28-s + 102·29-s − 265·31-s − 32·32-s − 252·34-s − 65·37-s − 58·38-s + 240·41-s + 367·43-s + 48·44-s + 36·46-s + 72·47-s + 18·49-s − 200·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.02·7-s − 0.353·8-s + 0.328·11-s − 1.06·13-s − 0.725·14-s + 1/4·16-s + 1.79·17-s + 0.350·19-s − 0.232·22-s − 0.163·23-s + 0.754·26-s + 0.512·28-s + 0.653·29-s − 1.53·31-s − 0.176·32-s − 1.27·34-s − 0.288·37-s − 0.247·38-s + 0.914·41-s + 1.30·43-s + 0.164·44-s + 0.115·46-s + 0.223·47-s + 0.0524·49-s − 0.533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.808684625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.808684625\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 29 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 265 T + p^{3} T^{2} \) |
| 37 | \( 1 + 65 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 367 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 102 T + p^{3} T^{2} \) |
| 61 | \( 1 + 103 T + p^{3} T^{2} \) |
| 67 | \( 1 - 52 T + p^{3} T^{2} \) |
| 71 | \( 1 + 582 T + p^{3} T^{2} \) |
| 73 | \( 1 + 65 T + p^{3} T^{2} \) |
| 79 | \( 1 - 173 T + p^{3} T^{2} \) |
| 83 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + 822 T + p^{3} T^{2} \) |
| 97 | \( 1 + 821 T + p^{3} T^{2} \) |
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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
−9.261513879645337400933412324940, −8.406975625085668475259073413382, −7.55687771413786601153758532743, −7.23671279382427447072684022328, −5.84258021680306488073423505542, −5.19869959904973264969567044957, −4.06923404796788421568586835373, −2.86471949604792429651164777776, −1.76013483454362170662338470937, −0.77950358126912367740675449275,
0.77950358126912367740675449275, 1.76013483454362170662338470937, 2.86471949604792429651164777776, 4.06923404796788421568586835373, 5.19869959904973264969567044957, 5.84258021680306488073423505542, 7.23671279382427447072684022328, 7.55687771413786601153758532743, 8.406975625085668475259073413382, 9.261513879645337400933412324940
