L(s) = 1 | − 3-s − 5-s − 4.78·7-s + 9-s − 4.10·11-s − 1.27·13-s + 15-s + 1.85·17-s − 6.92·19-s + 4.78·21-s − 2.96·23-s + 25-s − 27-s − 3.82·29-s − 5.62·31-s + 4.10·33-s + 4.78·35-s − 8.61·37-s + 1.27·39-s + 0.308·41-s − 0.994·43-s − 45-s + 5.62·47-s + 15.9·49-s − 1.85·51-s − 1.68·53-s + 4.10·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.81·7-s + 0.333·9-s − 1.23·11-s − 0.353·13-s + 0.258·15-s + 0.450·17-s − 1.58·19-s + 1.04·21-s − 0.617·23-s + 0.200·25-s − 0.192·27-s − 0.710·29-s − 1.00·31-s + 0.715·33-s + 0.809·35-s − 1.41·37-s + 0.204·39-s + 0.0482·41-s − 0.151·43-s − 0.149·45-s + 0.820·47-s + 2.27·49-s − 0.260·51-s − 0.231·53-s + 0.553·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1483293195\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1483293195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 + 5.62T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 - 0.308T + 41T^{2} \) |
| 43 | \( 1 + 0.994T + 43T^{2} \) |
| 47 | \( 1 - 5.62T + 47T^{2} \) |
| 53 | \( 1 + 1.68T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 7.31T + 89T^{2} \) |
| 97 | \( 1 + 3.20T + 97T^{2} \) |
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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
−8.482522417892719435133391821961, −7.46599900388839803900901256214, −7.05808746536438609224685238717, −6.10567590122992176388991846443, −5.69504127705379214149076198405, −4.67895837979613219740934149453, −3.78332671669224753064924167408, −3.06772142907504727712472574836, −2.06240197009087050248387806225, −0.20940619156881488401705517987,
0.20940619156881488401705517987, 2.06240197009087050248387806225, 3.06772142907504727712472574836, 3.78332671669224753064924167408, 4.67895837979613219740934149453, 5.69504127705379214149076198405, 6.10567590122992176388991846443, 7.05808746536438609224685238717, 7.46599900388839803900901256214, 8.482522417892719435133391821961