| L(s) = 1 | − 0.302·3-s + 0.697·7-s − 2.90·9-s + 11-s + 1.60·13-s + 4.30·17-s − 19-s − 0.211·21-s + 3.90·23-s + 1.78·27-s − 1.69·29-s + 1.60·31-s − 0.302·33-s + 7.60·37-s − 0.486·39-s + 8.21·41-s + 7.21·43-s + 5.60·47-s − 6.51·49-s − 1.30·51-s − 6.51·53-s + 0.302·57-s − 5.60·59-s + 3.30·61-s − 2.02·63-s + 8·67-s − 1.18·69-s + ⋯ |
| L(s) = 1 | − 0.174·3-s + 0.263·7-s − 0.969·9-s + 0.301·11-s + 0.445·13-s + 1.04·17-s − 0.229·19-s − 0.0460·21-s + 0.814·23-s + 0.344·27-s − 0.315·29-s + 0.288·31-s − 0.0527·33-s + 1.25·37-s − 0.0778·39-s + 1.28·41-s + 1.09·43-s + 0.817·47-s − 0.930·49-s − 0.182·51-s − 0.894·53-s + 0.0401·57-s − 0.729·59-s + 0.422·61-s − 0.255·63-s + 0.977·67-s − 0.142·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542079132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542079132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 13 | \( 1 - 1.60T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 + 6.51T + 53T^{2} \) |
| 59 | \( 1 + 5.60T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 + 1.90T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 - 6.90T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 + 7.11T + 97T^{2} \) |
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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
−9.752690828041502805500983934160, −9.054439949055151879200586179072, −8.193846260137039119208631568290, −7.48421885182019267138187204909, −6.30722966550186665045730844819, −5.69778851114034774892655963277, −4.71139223992248218188987295726, −3.58187718699789786972620902069, −2.55413362171532799754577225862, −0.994557187738439722557198564547,
0.994557187738439722557198564547, 2.55413362171532799754577225862, 3.58187718699789786972620902069, 4.71139223992248218188987295726, 5.69778851114034774892655963277, 6.30722966550186665045730844819, 7.48421885182019267138187204909, 8.193846260137039119208631568290, 9.054439949055151879200586179072, 9.752690828041502805500983934160
