L(s) = 1 | − 0.585·5-s + 3.69·7-s − 4.14·11-s + 3.41·13-s − 2.82·17-s − 6.30·19-s + 6.75·23-s − 4.65·25-s − 7.41·29-s + 3.06·31-s − 2.16·35-s + 9.07·37-s + 4·41-s − 1.08·43-s − 3.06·47-s + 6.65·49-s − 4.58·53-s + 2.42·55-s − 1.08·59-s − 1.07·61-s − 2·65-s + 1.97·67-s + 8.02·71-s − 6.48·73-s − 15.3·77-s − 14.7·79-s − 13.6·83-s + ⋯ |
L(s) = 1 | − 0.261·5-s + 1.39·7-s − 1.24·11-s + 0.946·13-s − 0.685·17-s − 1.44·19-s + 1.40·23-s − 0.931·25-s − 1.37·29-s + 0.549·31-s − 0.365·35-s + 1.49·37-s + 0.624·41-s − 0.165·43-s − 0.446·47-s + 0.950·49-s − 0.629·53-s + 0.327·55-s − 0.140·59-s − 0.137·61-s − 0.248·65-s + 0.241·67-s + 0.952·71-s − 0.759·73-s − 1.74·77-s − 1.66·79-s − 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 + 7.41T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 + 1.08T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−7.57215006728486142387231879555, −6.74296565437651285290087324360, −5.90165196871777992755349370870, −5.30831334700932836176600640147, −4.47805994756070011413855338964, −4.10764818238172151577803702751, −2.92469802377438730878936135794, −2.15789399517423854871645815589, −1.32222178361332820850877716905, 0,
1.32222178361332820850877716905, 2.15789399517423854871645815589, 2.92469802377438730878936135794, 4.10764818238172151577803702751, 4.47805994756070011413855338964, 5.30831334700932836176600640147, 5.90165196871777992755349370870, 6.74296565437651285290087324360, 7.57215006728486142387231879555