L(s) = 1 | + 5-s + 7-s − 2·13-s + 6·19-s − 23-s + 25-s + 6·29-s + 35-s − 8·37-s − 6·41-s − 10·43-s + 2·47-s + 49-s − 2·53-s − 4·59-s + 8·61-s − 2·65-s − 2·67-s − 10·71-s + 6·73-s − 4·83-s + 10·89-s − 2·91-s + 6·95-s − 6·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.554·13-s + 1.37·19-s − 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s − 1.31·37-s − 0.937·41-s − 1.52·43-s + 0.291·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s + 1.02·61-s − 0.248·65-s − 0.244·67-s − 1.18·71-s + 0.702·73-s − 0.439·83-s + 1.05·89-s − 0.209·91-s + 0.615·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.74290714282602, −13.56795776790608, −12.91309783991488, −12.25893888979287, −11.87183509139860, −11.64339488157849, −10.80493363938743, −10.44662579801857, −9.824238291876279, −9.655164419697745, −8.864946350897788, −8.459123474611340, −7.957649546527444, −7.300661811203346, −6.921124148566970, −6.371146405441809, −5.696188298954361, −5.120717917327116, −4.903354837532737, −4.153402664201630, −3.338745611682537, −3.012601289240048, −2.179037577070897, −1.623320751113510, −0.9597139478648421, 0,
0.9597139478648421, 1.623320751113510, 2.179037577070897, 3.012601289240048, 3.338745611682537, 4.153402664201630, 4.903354837532737, 5.120717917327116, 5.696188298954361, 6.371146405441809, 6.921124148566970, 7.300661811203346, 7.957649546527444, 8.459123474611340, 8.864946350897788, 9.655164419697745, 9.824238291876279, 10.44662579801857, 10.80493363938743, 11.64339488157849, 11.87183509139860, 12.25893888979287, 12.91309783991488, 13.56795776790608, 13.74290714282602