L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 2·13-s − 2·14-s + 16-s − 6·17-s + 2·19-s − 20-s + 25-s − 2·26-s + 2·28-s − 6·29-s − 10·31-s − 32-s + 6·34-s − 2·35-s + 37-s − 2·38-s + 40-s + 6·41-s − 4·43-s + 6·47-s − 3·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s + 0.164·37-s − 0.324·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.273671623566626426515406948175, −7.59731631879469280117277624063, −7.01389829611117685755868614566, −6.09945696108452207540402656878, −5.25330564043371456613299060211, −4.31005201030842057232591337712, −3.49870387163770249696581998319, −2.29716886023620970672769162796, −1.42055400977912848245367112580, 0,
1.42055400977912848245367112580, 2.29716886023620970672769162796, 3.49870387163770249696581998319, 4.31005201030842057232591337712, 5.25330564043371456613299060211, 6.09945696108452207540402656878, 7.01389829611117685755868614566, 7.59731631879469280117277624063, 8.273671623566626426515406948175