| L(s) = 1 | + 1.39·3-s + 1.17·5-s + 7-s − 1.05·9-s − 0.0631·11-s + 1.63·15-s + 5.07·17-s + 3.27·19-s + 1.39·21-s + 9.35·23-s − 3.62·25-s − 5.65·27-s + 3.49·29-s + 4.50·31-s − 0.0882·33-s + 1.17·35-s + 9.59·37-s − 8.38·41-s − 11.0·43-s − 1.23·45-s + 4.08·47-s + 49-s + 7.08·51-s + 1.39·53-s − 0.0741·55-s + 4.56·57-s − 4.76·59-s + ⋯ |
| L(s) = 1 | + 0.806·3-s + 0.525·5-s + 0.377·7-s − 0.350·9-s − 0.0190·11-s + 0.423·15-s + 1.23·17-s + 0.750·19-s + 0.304·21-s + 1.94·23-s − 0.724·25-s − 1.08·27-s + 0.648·29-s + 0.809·31-s − 0.0153·33-s + 0.198·35-s + 1.57·37-s − 1.30·41-s − 1.68·43-s − 0.183·45-s + 0.596·47-s + 0.142·49-s + 0.992·51-s + 0.191·53-s − 0.0100·55-s + 0.605·57-s − 0.620·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.594754607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.594754607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 11 | \( 1 + 0.0631T + 11T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 9.35T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 9.59T + 37T^{2} \) |
| 41 | \( 1 + 8.38T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 1.39T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 2.15T + 73T^{2} \) |
| 79 | \( 1 - 17.7T + 79T^{2} \) |
| 83 | \( 1 + 9.36T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.928927447556744590897026407824, −7.09362433127515628989778927462, −6.38066703557045212337283261979, −5.46330108033924086295735900607, −5.13534939954984906199754278717, −4.11310216043387769092360845645, −3.11696711670881720341494379618, −2.84947731016054190343373487585, −1.76696119525466527432033522043, −0.913342428546967808376663630924,
0.913342428546967808376663630924, 1.76696119525466527432033522043, 2.84947731016054190343373487585, 3.11696711670881720341494379618, 4.11310216043387769092360845645, 5.13534939954984906199754278717, 5.46330108033924086295735900607, 6.38066703557045212337283261979, 7.09362433127515628989778927462, 7.928927447556744590897026407824
