| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 11-s + 2·13-s − 4·14-s − 16-s − 2·17-s − 22-s + 8·23-s + 2·26-s + 4·28-s + 6·29-s − 8·31-s + 5·32-s − 2·34-s − 6·37-s + 2·41-s + 44-s + 8·46-s + 8·47-s + 9·49-s − 2·52-s + 6·53-s + 12·56-s + 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 0.301·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.213·22-s + 1.66·23-s + 0.392·26-s + 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s − 0.986·37-s + 0.312·41-s + 0.150·44-s + 1.17·46-s + 1.16·47-s + 9/7·49-s − 0.277·52-s + 0.824·53-s + 1.60·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417322571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417322571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.052312079360069985753137252747, −8.360742840269281397590237858671, −7.06746602683221101194508851954, −6.57667809882336453792833363635, −5.68297890149735059120081895574, −5.05334108529123054060716186457, −3.97182275742999473711448669060, −3.37409785551698507568515532805, −2.55490898649220854449117581353, −0.66299652490358276934219865411,
0.66299652490358276934219865411, 2.55490898649220854449117581353, 3.37409785551698507568515532805, 3.97182275742999473711448669060, 5.05334108529123054060716186457, 5.68297890149735059120081895574, 6.57667809882336453792833363635, 7.06746602683221101194508851954, 8.360742840269281397590237858671, 9.052312079360069985753137252747
