L(s) = 1 | + 3-s − 7-s + 9-s − 5·11-s − 2·13-s − 4·17-s + 19-s − 21-s − 23-s + 27-s − 4·29-s + 2·31-s − 5·33-s + 10·37-s − 2·39-s + 5·41-s − 8·43-s + 7·47-s + 49-s − 4·51-s − 9·53-s + 57-s + 4·59-s + 10·61-s − 63-s − 8·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.870·33-s + 1.64·37-s − 0.320·39-s + 0.780·41-s − 1.21·43-s + 1.02·47-s + 1/7·49-s − 0.560·51-s − 1.23·53-s + 0.132·57-s + 0.520·59-s + 1.28·61-s − 0.125·63-s − 0.977·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9389330484\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9389330484\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.50152102038595, −13.37709121714546, −12.97520151030274, −12.47347922889907, −11.86957287871134, −11.24912986076041, −10.84459072838619, −10.22084440318814, −9.827889979358413, −9.392721193941437, −8.781726790425449, −8.255886126266111, −7.794700012419564, −7.288998291481473, −6.880023753878999, −6.061163373302228, −5.657281525277958, −4.975541929780832, −4.402052391774419, −3.959498288650867, −2.976454835902775, −2.738910718173420, −2.180929126410355, −1.355921106322460, −0.2843234748651964,
0.2843234748651964, 1.355921106322460, 2.180929126410355, 2.738910718173420, 2.976454835902775, 3.959498288650867, 4.402052391774419, 4.975541929780832, 5.657281525277958, 6.061163373302228, 6.880023753878999, 7.288998291481473, 7.794700012419564, 8.255886126266111, 8.781726790425449, 9.392721193941437, 9.827889979358413, 10.22084440318814, 10.84459072838619, 11.24912986076041, 11.86957287871134, 12.47347922889907, 12.97520151030274, 13.37709121714546, 13.50152102038595