L(s) = 1 | + 2-s − 0.00179·3-s + 4-s − 2.84·5-s − 0.00179·6-s + 2.42·7-s + 8-s − 2.99·9-s − 2.84·10-s − 4.79·11-s − 0.00179·12-s + 0.240·13-s + 2.42·14-s + 0.00510·15-s + 16-s + 0.878·17-s − 2.99·18-s − 19-s − 2.84·20-s − 0.00434·21-s − 4.79·22-s − 5.60·23-s − 0.00179·24-s + 3.09·25-s + 0.240·26-s + 0.0107·27-s + 2.42·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.00103·3-s + 0.5·4-s − 1.27·5-s − 0.000733·6-s + 0.914·7-s + 0.353·8-s − 0.999·9-s − 0.899·10-s − 1.44·11-s − 0.000518·12-s + 0.0665·13-s + 0.646·14-s + 0.00131·15-s + 0.250·16-s + 0.213·17-s − 0.707·18-s − 0.229·19-s − 0.636·20-s − 0.000948·21-s − 1.02·22-s − 1.16·23-s − 0.000366·24-s + 0.619·25-s + 0.0470·26-s + 0.00207·27-s + 0.457·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.770800838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770800838\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 0.00179T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 - 0.240T + 13T^{2} \) |
| 17 | \( 1 - 0.878T + 17T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 - 8.02T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 8.34T + 53T^{2} \) |
| 59 | \( 1 - 5.86T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 4.81T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.74425297875422857832956539802, −7.50342620543815144738900880863, −6.13902510794868997222831285879, −5.79512155580896013778395910896, −4.76880398094872035117917659450, −4.49610244426549352727180994248, −3.53642940829767121123746693562, −2.81503977430265769689839252711, −2.05827387994533784595341699250, −0.56275310017068733770476228460,
0.56275310017068733770476228460, 2.05827387994533784595341699250, 2.81503977430265769689839252711, 3.53642940829767121123746693562, 4.49610244426549352727180994248, 4.76880398094872035117917659450, 5.79512155580896013778395910896, 6.13902510794868997222831285879, 7.50342620543815144738900880863, 7.74425297875422857832956539802