| L(s) = 1 | + 8.94·3-s − 8·4-s + 53.0·9-s + 72·11-s − 71.5·12-s + 40.2·13-s + 64·16-s − 102.·17-s + 232.·27-s + 54·29-s + 643.·33-s − 424.·36-s + 360.·39-s − 576·44-s + 178.·47-s + 572.·48-s − 920·51-s − 321.·52-s − 512·64-s + 822.·68-s + 828·71-s − 523.·73-s + 236·79-s + 649.·81-s + 1.51e3·83-s + 482.·87-s + 1.65e3·97-s + ⋯ |
| L(s) = 1 | + 1.72·3-s − 4-s + 1.96·9-s + 1.97·11-s − 1.72·12-s + 0.858·13-s + 16-s − 1.46·17-s + 1.65·27-s + 0.345·29-s + 3.39·33-s − 1.96·36-s + 1.47·39-s − 1.97·44-s + 0.555·47-s + 1.72·48-s − 2.52·51-s − 0.858·52-s − 64-s + 1.46·68-s + 1.38·71-s − 0.838·73-s + 0.336·79-s + 0.890·81-s + 1.99·83-s + 0.595·87-s + 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.903011669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.903011669\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 8T^{2} \) |
| 3 | \( 1 - 8.94T + 27T^{2} \) |
| 11 | \( 1 - 72T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 - 178.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 - 828T + 3.57e5T^{2} \) |
| 73 | \( 1 + 523.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 236T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.51e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−9.090135283526067078350866291031, −8.769904156677037160773649183831, −8.082877394865753694625839501527, −6.99813106634805990282708606036, −6.19151756371168909903851090030, −4.61587083825603573724369320581, −3.96181066218513057275731020960, −3.38907776548369878692592967985, −2.04081861390040160044146539731, −1.00980507965976575891081860968,
1.00980507965976575891081860968, 2.04081861390040160044146539731, 3.38907776548369878692592967985, 3.96181066218513057275731020960, 4.61587083825603573724369320581, 6.19151756371168909903851090030, 6.99813106634805990282708606036, 8.082877394865753694625839501527, 8.769904156677037160773649183831, 9.090135283526067078350866291031
