L(s) = 1 | + 3-s − 5-s − 4.38·7-s + 9-s − 6.38·11-s − 1.13·13-s − 15-s + 1.72·17-s − 19-s − 4.38·21-s − 1.52·23-s + 25-s + 27-s + 6.65·29-s + 3.72·31-s − 6.38·33-s + 4.38·35-s − 4.59·37-s − 1.13·39-s − 9.64·41-s − 2.59·43-s − 45-s + 9.52·47-s + 12.2·49-s + 1.72·51-s + 11.5·53-s + 6.38·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.65·7-s + 0.333·9-s − 1.92·11-s − 0.314·13-s − 0.258·15-s + 0.419·17-s − 0.229·19-s − 0.957·21-s − 0.317·23-s + 0.200·25-s + 0.192·27-s + 1.23·29-s + 0.669·31-s − 1.11·33-s + 0.741·35-s − 0.755·37-s − 0.181·39-s − 1.50·41-s − 0.395·43-s − 0.149·45-s + 1.38·47-s + 1.75·49-s + 0.242·51-s + 1.58·53-s + 0.861·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.005637441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005637441\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4.38T + 7T^{2} \) |
| 11 | \( 1 + 6.38T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 - 3.72T + 31T^{2} \) |
| 37 | \( 1 + 4.59T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 1.79T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 + 8.50T + 73T^{2} \) |
| 79 | \( 1 - 9.25T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 4.59T + 97T^{2} \) |
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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
−8.330756150970532981866916109783, −7.61679748027876702263386187870, −6.99859155621159722215605466215, −6.23820889168846001023862730722, −5.36487259819267208050322338016, −4.56101495918939957952013174343, −3.49469238893022622688385567644, −2.98219200991965002596393503180, −2.27699329572006245533550470561, −0.50618660957208422987682212753,
0.50618660957208422987682212753, 2.27699329572006245533550470561, 2.98219200991965002596393503180, 3.49469238893022622688385567644, 4.56101495918939957952013174343, 5.36487259819267208050322338016, 6.23820889168846001023862730722, 6.99859155621159722215605466215, 7.61679748027876702263386187870, 8.330756150970532981866916109783