| L(s) = 1 | − 3-s + 9-s − 5·11-s + 2·13-s − 6·17-s + 2·19-s + 5·23-s − 27-s − 5·29-s + 4·31-s + 5·33-s + 37-s − 2·39-s + 12·41-s + 5·43-s − 2·47-s + 6·51-s + 14·53-s − 2·57-s − 2·59-s − 5·67-s − 5·69-s − 9·71-s − 10·73-s + 11·79-s + 81-s − 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.04·23-s − 0.192·27-s − 0.928·29-s + 0.718·31-s + 0.870·33-s + 0.164·37-s − 0.320·39-s + 1.87·41-s + 0.762·43-s − 0.291·47-s + 0.840·51-s + 1.92·53-s − 0.264·57-s − 0.260·59-s − 0.610·67-s − 0.601·69-s − 1.06·71-s − 1.17·73-s + 1.23·79-s + 1/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196061547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196061547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.24537694966575, −14.83186894270435, −13.95893667793070, −13.36403064229581, −13.09015202676048, −12.67754913686674, −11.84461511539045, −11.35526038185878, −10.79953027621855, −10.57319819220491, −9.842640195320540, −9.091039548713014, −8.765083165448455, −7.908713229155782, −7.432785311895164, −6.912141094722261, −6.114276960139483, −5.678785012536910, −5.045965226995499, −4.453850590821810, −3.838552327340612, −2.783488634832731, −2.434161635629072, −1.342953823389855, −0.4542415134034197,
0.4542415134034197, 1.342953823389855, 2.434161635629072, 2.783488634832731, 3.838552327340612, 4.453850590821810, 5.045965226995499, 5.678785012536910, 6.114276960139483, 6.912141094722261, 7.432785311895164, 7.908713229155782, 8.765083165448455, 9.091039548713014, 9.842640195320540, 10.57319819220491, 10.79953027621855, 11.35526038185878, 11.84461511539045, 12.67754913686674, 13.09015202676048, 13.36403064229581, 13.95893667793070, 14.83186894270435, 15.24537694966575
