| L(s) = 1 | + 1.65·3-s + 1.16·5-s + 5.22·7-s − 0.261·9-s + 1.87·11-s + 3.59·13-s + 1.93·15-s − 4.39·17-s + 3.70·19-s + 8.64·21-s − 7.20·23-s − 3.63·25-s − 5.39·27-s + 5.84·29-s + 1.35·31-s + 3.10·33-s + 6.10·35-s + 3.71·37-s + 5.94·39-s − 2.17·41-s − 10.7·43-s − 0.305·45-s − 2.70·47-s + 20.2·49-s − 7.27·51-s + 13.6·53-s + 2.19·55-s + ⋯ |
| L(s) = 1 | + 0.955·3-s + 0.522·5-s + 1.97·7-s − 0.0870·9-s + 0.565·11-s + 0.996·13-s + 0.499·15-s − 1.06·17-s + 0.849·19-s + 1.88·21-s − 1.50·23-s − 0.726·25-s − 1.03·27-s + 1.08·29-s + 0.243·31-s + 0.540·33-s + 1.03·35-s + 0.610·37-s + 0.952·39-s − 0.340·41-s − 1.64·43-s − 0.0455·45-s − 0.394·47-s + 2.89·49-s − 1.01·51-s + 1.87·53-s + 0.295·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.500263442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.500263442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 7 | \( 1 - 5.22T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 3.71T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 2.91T + 67T^{2} \) |
| 71 | \( 1 - 5.23T + 71T^{2} \) |
| 73 | \( 1 + 6.15T + 73T^{2} \) |
| 79 | \( 1 - 8.25T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 8.59T + 97T^{2} \) |
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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
−8.204879741638843655122702759305, −7.31750816592898812750743907806, −6.44250392591118466266401066814, −5.69384785253866673982913446803, −5.01909264232602321800842163361, −4.15061717497890052323549716976, −3.62906478174332025167073356275, −2.35772515137122977528838209862, −1.96848588109068147642518495697, −1.08163607615291862735836011752,
1.08163607615291862735836011752, 1.96848588109068147642518495697, 2.35772515137122977528838209862, 3.62906478174332025167073356275, 4.15061717497890052323549716976, 5.01909264232602321800842163361, 5.69384785253866673982913446803, 6.44250392591118466266401066814, 7.31750816592898812750743907806, 8.204879741638843655122702759305
