| L(s) = 1 | + 0.780·3-s + 0.801·5-s − 4.30·7-s − 2.39·9-s + 11-s + 0.708·13-s + 0.625·15-s − 4.92·17-s − 4.78·19-s − 3.36·21-s + 6.77·23-s − 4.35·25-s − 4.20·27-s − 0.762·29-s − 0.430·31-s + 0.780·33-s − 3.45·35-s + 1.04·37-s + 0.553·39-s − 0.926·41-s + 6.17·43-s − 1.91·45-s + 4.77·47-s + 11.5·49-s − 3.84·51-s + 13.0·53-s + 0.801·55-s + ⋯ |
| L(s) = 1 | + 0.450·3-s + 0.358·5-s − 1.62·7-s − 0.796·9-s + 0.301·11-s + 0.196·13-s + 0.161·15-s − 1.19·17-s − 1.09·19-s − 0.733·21-s + 1.41·23-s − 0.871·25-s − 0.809·27-s − 0.141·29-s − 0.0773·31-s + 0.135·33-s − 0.583·35-s + 0.172·37-s + 0.0885·39-s − 0.144·41-s + 0.941·43-s − 0.285·45-s + 0.697·47-s + 1.64·49-s − 0.538·51-s + 1.78·53-s + 0.108·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362605150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362605150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 0.780T + 3T^{2} \) |
| 5 | \( 1 - 0.801T + 5T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 13 | \( 1 - 0.708T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 4.78T + 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 + 0.762T + 29T^{2} \) |
| 31 | \( 1 + 0.430T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 + 0.926T + 41T^{2} \) |
| 43 | \( 1 - 6.17T + 43T^{2} \) |
| 47 | \( 1 - 4.77T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 0.761T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 4.75T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 8.86T + 83T^{2} \) |
| 89 | \( 1 + 0.640T + 89T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
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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
−8.297709269402186619271687810148, −7.16779321596843572529227270156, −6.65910280213990401721776769873, −6.04236172594616873354902134360, −5.40202941625827289583853736817, −4.18998505676943273965799941069, −3.60893396528277396077333319470, −2.69832301966748875401414816547, −2.18332918622719249607710734566, −0.56260563463424447181911589029,
0.56260563463424447181911589029, 2.18332918622719249607710734566, 2.69832301966748875401414816547, 3.60893396528277396077333319470, 4.18998505676943273965799941069, 5.40202941625827289583853736817, 6.04236172594616873354902134360, 6.65910280213990401721776769873, 7.16779321596843572529227270156, 8.297709269402186619271687810148
