L(s) = 1 | − 1.20·2-s − 0.933·3-s − 0.547·4-s + 1.12·6-s + 2.09·7-s + 3.07·8-s − 2.12·9-s − 4.33·11-s + 0.511·12-s − 1.99·13-s − 2.52·14-s − 2.60·16-s + 2.99·17-s + 2.56·18-s + 0.142·19-s − 1.95·21-s + 5.22·22-s − 0.193·23-s − 2.86·24-s + 2.40·26-s + 4.78·27-s − 1.14·28-s − 0.0799·29-s − 9.71·31-s − 3.00·32-s + 4.04·33-s − 3.60·34-s + ⋯ |
L(s) = 1 | − 0.852·2-s − 0.539·3-s − 0.273·4-s + 0.459·6-s + 0.791·7-s + 1.08·8-s − 0.709·9-s − 1.30·11-s + 0.147·12-s − 0.552·13-s − 0.674·14-s − 0.651·16-s + 0.726·17-s + 0.604·18-s + 0.0326·19-s − 0.426·21-s + 1.11·22-s − 0.0403·23-s − 0.585·24-s + 0.470·26-s + 0.921·27-s − 0.216·28-s − 0.0148·29-s − 1.74·31-s − 0.530·32-s + 0.704·33-s − 0.619·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4400034087\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4400034087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 3 | \( 1 + 0.933T + 3T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 - 0.142T + 19T^{2} \) |
| 23 | \( 1 + 0.193T + 23T^{2} \) |
| 29 | \( 1 + 0.0799T + 29T^{2} \) |
| 31 | \( 1 + 9.71T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 43 | \( 1 + 0.129T + 43T^{2} \) |
| 47 | \( 1 + 0.573T + 47T^{2} \) |
| 53 | \( 1 + 4.38T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 3.84T + 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 9.15T + 89T^{2} \) |
| 97 | \( 1 + 9.88T + 97T^{2} \) |
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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
−8.186810225342372163953877090356, −7.56312627092414385828919815807, −6.94394464818376378278897946421, −5.66653212848530844272320639804, −5.28027503279966576817755537867, −4.73330593760185313266720866712, −3.64760972433460254360699829786, −2.56377905308509374207196040532, −1.60876189026174703810930735390, −0.41327615005422044072505719951,
0.41327615005422044072505719951, 1.60876189026174703810930735390, 2.56377905308509374207196040532, 3.64760972433460254360699829786, 4.73330593760185313266720866712, 5.28027503279966576817755537867, 5.66653212848530844272320639804, 6.94394464818376378278897946421, 7.56312627092414385828919815807, 8.186810225342372163953877090356