| L(s) = 1 | + 2.56·3-s − 0.561·5-s − 2.56·7-s + 3.56·9-s − 5.12·11-s + 13-s − 1.43·15-s + 5.68·17-s + 5.12·19-s − 6.56·21-s − 8·23-s − 4.68·25-s + 1.43·27-s − 2·29-s + 4·31-s − 13.1·33-s + 1.43·35-s + 9.68·37-s + 2.56·39-s − 3.12·41-s + 5.43·43-s − 2.00·45-s − 0.315·47-s − 0.438·49-s + 14.5·51-s + 3.12·53-s + 2.87·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.251·5-s − 0.968·7-s + 1.18·9-s − 1.54·11-s + 0.277·13-s − 0.371·15-s + 1.37·17-s + 1.17·19-s − 1.43·21-s − 1.66·23-s − 0.936·25-s + 0.276·27-s − 0.371·29-s + 0.718·31-s − 2.28·33-s + 0.243·35-s + 1.59·37-s + 0.410·39-s − 0.487·41-s + 0.829·43-s − 0.298·45-s − 0.0459·47-s − 0.0626·49-s + 2.03·51-s + 0.428·53-s + 0.387·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330995200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330995200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 + 0.315T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−13.71753072328683946346817848159, −13.10195645373867280775639021405, −11.90591244910145441082720776956, −10.13223292975874914445498474652, −9.587407953010303947547664738248, −8.098504966471858535674901672523, −7.61495739064935920384504857216, −5.74788173249571342817361812177, −3.72784241969437990537506006035, −2.68088295258062156328134462169,
2.68088295258062156328134462169, 3.72784241969437990537506006035, 5.74788173249571342817361812177, 7.61495739064935920384504857216, 8.098504966471858535674901672523, 9.587407953010303947547664738248, 10.13223292975874914445498474652, 11.90591244910145441082720776956, 13.10195645373867280775639021405, 13.71753072328683946346817848159
