L(s) = 1 | + 0.0675·3-s − 1.58·5-s + 7-s − 2.99·9-s + 11-s − 13-s − 0.107·15-s − 7.43·17-s − 0.425·19-s + 0.0675·21-s + 2.83·23-s − 2.47·25-s − 0.405·27-s + 2.09·29-s + 9.18·31-s + 0.0675·33-s − 1.58·35-s + 3.75·37-s − 0.0675·39-s − 3.36·41-s − 7.73·43-s + 4.75·45-s + 9.56·47-s + 49-s − 0.502·51-s − 3.70·53-s − 1.58·55-s + ⋯ |
L(s) = 1 | + 0.0390·3-s − 0.710·5-s + 0.377·7-s − 0.998·9-s + 0.301·11-s − 0.277·13-s − 0.0277·15-s − 1.80·17-s − 0.0976·19-s + 0.0147·21-s + 0.591·23-s − 0.495·25-s − 0.0779·27-s + 0.389·29-s + 1.64·31-s + 0.0117·33-s − 0.268·35-s + 0.617·37-s − 0.0108·39-s − 0.526·41-s − 1.17·43-s + 0.709·45-s + 1.39·47-s + 0.142·49-s − 0.0703·51-s − 0.508·53-s − 0.214·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219337877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219337877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 0.0675T + 3T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 17 | \( 1 + 7.43T + 17T^{2} \) |
| 19 | \( 1 + 0.425T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 2.09T + 29T^{2} \) |
| 31 | \( 1 - 9.18T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 3.70T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 1.17T + 61T^{2} \) |
| 67 | \( 1 + 0.996T + 67T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 + 4.05T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.401259616060922584077962217392, −7.919791430353267366392802203365, −6.88059793859993284321943077009, −6.41611558349575755296717659148, −5.37903832373222249631241093256, −4.60702767797411899295015015292, −3.94884615160668274527250826973, −2.89796323006007384461581430595, −2.12211351824731037292838855785, −0.60909132930974332063683414057,
0.60909132930974332063683414057, 2.12211351824731037292838855785, 2.89796323006007384461581430595, 3.94884615160668274527250826973, 4.60702767797411899295015015292, 5.37903832373222249631241093256, 6.41611558349575755296717659148, 6.88059793859993284321943077009, 7.919791430353267366392802203365, 8.401259616060922584077962217392