| L(s) = 1 | + 2.56·3-s − 3.20·7-s + 3.56·9-s − 1.46·11-s − 13-s − 5.10·17-s + 1.74·19-s − 8.21·21-s + 9.22·23-s + 1.43·27-s + 9.86·29-s + 7.00·31-s − 3.74·33-s + 3.12·37-s − 2.56·39-s + 4.46·41-s + 1.01·43-s − 4.12·47-s + 3.27·49-s − 13.0·51-s − 2.54·53-s + 4.46·57-s − 5.40·59-s + 0.543·61-s − 11.4·63-s + 12.7·67-s + 23.6·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 1.21·7-s + 1.18·9-s − 0.440·11-s − 0.277·13-s − 1.23·17-s + 0.400·19-s − 1.79·21-s + 1.92·23-s + 0.276·27-s + 1.83·29-s + 1.25·31-s − 0.651·33-s + 0.513·37-s − 0.410·39-s + 0.697·41-s + 0.155·43-s − 0.602·47-s + 0.467·49-s − 1.83·51-s − 0.349·53-s + 0.591·57-s − 0.703·59-s + 0.0696·61-s − 1.43·63-s + 1.55·67-s + 2.84·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.827739436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827739436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 - 9.22T + 23T^{2} \) |
| 29 | \( 1 - 9.86T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 - 0.543T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 9.22T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 6.21T + 97T^{2} \) |
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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
−8.241287321630191116413100190787, −7.66121599148857462903942341930, −6.69294178615063481049290246439, −6.46559828677760409268761082989, −5.09316780651186380520527953967, −4.41530233983438993932617531913, −3.38738896617077713209228939910, −2.85191634883019743425068841195, −2.33180929550583545476913987344, −0.833042953231188714321806020150,
0.833042953231188714321806020150, 2.33180929550583545476913987344, 2.85191634883019743425068841195, 3.38738896617077713209228939910, 4.41530233983438993932617531913, 5.09316780651186380520527953967, 6.46559828677760409268761082989, 6.69294178615063481049290246439, 7.66121599148857462903942341930, 8.241287321630191116413100190787
