| L(s) = 1 | − 2.96·3-s − 3.39·5-s + 0.580·7-s + 5.81·9-s − 4.41·11-s + 6.41·13-s + 10.0·15-s + 1.13·17-s + 4.81·19-s − 1.72·21-s − 3.29·23-s + 6.55·25-s − 8.36·27-s + 6.69·29-s + 5.07·31-s + 13.1·33-s − 1.97·35-s − 11.1·37-s − 19.0·39-s + 10.0·41-s − 43-s − 19.7·45-s − 0.449·47-s − 6.66·49-s − 3.37·51-s + 0.385·53-s + 14.9·55-s + ⋯ |
| L(s) = 1 | − 1.71·3-s − 1.51·5-s + 0.219·7-s + 1.93·9-s − 1.33·11-s + 1.77·13-s + 2.60·15-s + 0.276·17-s + 1.10·19-s − 0.376·21-s − 0.686·23-s + 1.31·25-s − 1.61·27-s + 1.24·29-s + 0.911·31-s + 2.28·33-s − 0.333·35-s − 1.83·37-s − 3.04·39-s + 1.57·41-s − 0.152·43-s − 2.94·45-s − 0.0656·47-s − 0.951·49-s − 0.473·51-s + 0.0528·53-s + 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5378657874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5378657874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 - 0.580T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 - 6.41T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 47 | \( 1 + 0.449T + 47T^{2} \) |
| 53 | \( 1 - 0.385T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 - 1.11T + 61T^{2} \) |
| 67 | \( 1 - 9.27T + 67T^{2} \) |
| 71 | \( 1 - 5.12T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 2.60T + 83T^{2} \) |
| 89 | \( 1 + 0.304T + 89T^{2} \) |
| 97 | \( 1 - 8.98T + 97T^{2} \) |
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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
−11.51305949204242683822957826400, −10.85964814398737986959734141557, −10.16481488444051672040528015234, −8.400745174119402433277768831414, −7.69209901050556545818181407150, −6.60593657715634277300521646068, −5.56103729448492203836318650600, −4.66903997282954527127512924276, −3.52694971229213187538204009269, −0.799854793364524765552904086986,
0.799854793364524765552904086986, 3.52694971229213187538204009269, 4.66903997282954527127512924276, 5.56103729448492203836318650600, 6.60593657715634277300521646068, 7.69209901050556545818181407150, 8.400745174119402433277768831414, 10.16481488444051672040528015234, 10.85964814398737986959734141557, 11.51305949204242683822957826400
