L(s) = 1 | − 2·7-s − 4·11-s − 13-s + 6·17-s + 6·19-s − 5·25-s − 2·29-s + 6·31-s − 10·37-s − 8·41-s + 12·43-s + 12·47-s − 3·49-s − 6·53-s − 2·61-s + 2·67-s − 8·71-s + 14·73-s + 8·77-s − 4·79-s − 8·83-s − 4·89-s + 2·91-s + 14·97-s − 18·101-s − 4·103-s + 4·107-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 1.20·11-s − 0.277·13-s + 1.45·17-s + 1.37·19-s − 25-s − 0.371·29-s + 1.07·31-s − 1.64·37-s − 1.24·41-s + 1.82·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.256·61-s + 0.244·67-s − 0.949·71-s + 1.63·73-s + 0.911·77-s − 0.450·79-s − 0.878·83-s − 0.423·89-s + 0.209·91-s + 1.42·97-s − 1.79·101-s − 0.394·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.57823410697303097169540121767, −7.00907429585981817216868413060, −5.97097316442773123261763624730, −5.50877829531281648944449050521, −4.87246436998902955438509746746, −3.73436134603235551748934738311, −3.15306849273800822642792977376, −2.41737772174342246575260162871, −1.18710091181790114220093039148, 0,
1.18710091181790114220093039148, 2.41737772174342246575260162871, 3.15306849273800822642792977376, 3.73436134603235551748934738311, 4.87246436998902955438509746746, 5.50877829531281648944449050521, 5.97097316442773123261763624730, 7.00907429585981817216868413060, 7.57823410697303097169540121767