L(s) = 1 | + 7-s + 4·13-s − 6·17-s − 2·19-s − 5·25-s − 6·29-s − 4·31-s − 2·37-s − 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 6·59-s − 8·61-s + 4·67-s + 2·73-s + 8·79-s − 6·83-s + 6·89-s + 4·91-s − 10·97-s − 4·103-s + 12·107-s − 2·109-s − 6·113-s − 6·119-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.781·59-s − 1.02·61-s + 0.488·67-s + 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.635·89-s + 0.419·91-s − 1.01·97-s − 0.394·103-s + 1.16·107-s − 0.191·109-s − 0.564·113-s − 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.156759517304686892380266927456, −7.34808391035356365236166309157, −6.58338612370338623267809377565, −5.89372115656638311788103577429, −5.10276803747253805923263658121, −4.14318817559479393176326791421, −3.59991440544977347073267949607, −2.31147598100149017773134041140, −1.55194931839012879683964780078, 0,
1.55194931839012879683964780078, 2.31147598100149017773134041140, 3.59991440544977347073267949607, 4.14318817559479393176326791421, 5.10276803747253805923263658121, 5.89372115656638311788103577429, 6.58338612370338623267809377565, 7.34808391035356365236166309157, 8.156759517304686892380266927456