| L(s) = 1 | − 2-s − 1.67·3-s + 4-s + 1.67·6-s + 1.80·7-s − 8-s − 0.193·9-s − 6.44·11-s − 1.67·12-s − 1.80·14-s + 16-s − 0.481·17-s + 0.193·18-s + 6.28·19-s − 3.02·21-s + 6.44·22-s − 7.11·23-s + 1.67·24-s + 5.35·27-s + 1.80·28-s + 2.31·29-s − 3.25·31-s − 32-s + 10.7·33-s + 0.481·34-s − 0.193·36-s + 3.06·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.967·3-s + 0.5·4-s + 0.683·6-s + 0.682·7-s − 0.353·8-s − 0.0646·9-s − 1.94·11-s − 0.483·12-s − 0.482·14-s + 0.250·16-s − 0.116·17-s + 0.0457·18-s + 1.44·19-s − 0.660·21-s + 1.37·22-s − 1.48·23-s + 0.341·24-s + 1.02·27-s + 0.341·28-s + 0.429·29-s − 0.584·31-s − 0.176·32-s + 1.87·33-s + 0.0825·34-s − 0.0323·36-s + 0.503·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4780452802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4780452802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 17 | \( 1 + 0.481T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 0.906T + 53T^{2} \) |
| 59 | \( 1 + 6.57T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 0.649T + 67T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 + 2.60T + 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 1.15T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.893395106362837428268540449285, −7.29599501486909624796702646324, −6.34784589497958688186672494187, −5.66425509036892587932663429677, −5.21670787465782614003935748129, −4.56321768244380160702980194755, −3.27242434011752150917844370854, −2.51940274073992422630755727015, −1.55912956108049909702411675917, −0.39621005525475193402328147338,
0.39621005525475193402328147338, 1.55912956108049909702411675917, 2.51940274073992422630755727015, 3.27242434011752150917844370854, 4.56321768244380160702980194755, 5.21670787465782614003935748129, 5.66425509036892587932663429677, 6.34784589497958688186672494187, 7.29599501486909624796702646324, 7.893395106362837428268540449285
