| L(s) = 1 | − 0.199·3-s + 0.368·5-s + 1.63·7-s − 2.96·9-s − 0.834·11-s + 0.791·13-s − 0.0735·15-s − 7.69·17-s − 19-s − 0.326·21-s + 4.91·23-s − 4.86·25-s + 1.19·27-s + 7.77·29-s − 9.19·31-s + 0.166·33-s + 0.602·35-s + 6.82·37-s − 0.158·39-s − 5.55·41-s + 7.32·43-s − 1.08·45-s + 9.91·47-s − 4.32·49-s + 1.53·51-s − 0.543·53-s − 0.307·55-s + ⋯ |
| L(s) = 1 | − 0.115·3-s + 0.164·5-s + 0.618·7-s − 0.986·9-s − 0.251·11-s + 0.219·13-s − 0.0189·15-s − 1.86·17-s − 0.229·19-s − 0.0713·21-s + 1.02·23-s − 0.972·25-s + 0.229·27-s + 1.44·29-s − 1.65·31-s + 0.0290·33-s + 0.101·35-s + 1.12·37-s − 0.0253·39-s − 0.868·41-s + 1.11·43-s − 0.162·45-s + 1.44·47-s − 0.617·49-s + 0.215·51-s − 0.0746·53-s − 0.0414·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564836766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564836766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 0.199T + 3T^{2} \) |
| 5 | \( 1 - 0.368T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 0.834T + 11T^{2} \) |
| 13 | \( 1 - 0.791T + 13T^{2} \) |
| 17 | \( 1 + 7.69T + 17T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 + 5.55T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 9.91T + 47T^{2} \) |
| 53 | \( 1 + 0.543T + 53T^{2} \) |
| 59 | \( 1 - 8.81T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.46T + 73T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 0.948T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.156634018241262667457392771310, −7.39583334548267828184557710026, −6.58674387528022599930604459428, −5.96398174106690676382594663161, −5.18513630915983435707895210504, −4.55764186320858058508805712225, −3.68568801883070511110178028597, −2.59958300899948820132168364960, −2.02395127378664706044649900511, −0.64063471106590715623891723193,
0.64063471106590715623891723193, 2.02395127378664706044649900511, 2.59958300899948820132168364960, 3.68568801883070511110178028597, 4.55764186320858058508805712225, 5.18513630915983435707895210504, 5.96398174106690676382594663161, 6.58674387528022599930604459428, 7.39583334548267828184557710026, 8.156634018241262667457392771310
