L(s) = 1 | + 3.37·2-s + 3.40·4-s − 15.7·5-s + 17.1·7-s − 15.5·8-s − 53.0·10-s + 52.8·11-s + 57.9·14-s − 79.6·16-s + 71.0·17-s − 92.6·19-s − 53.5·20-s + 178.·22-s − 190.·23-s + 121.·25-s + 58.4·28-s + 128.·29-s − 3.29·31-s − 144.·32-s + 240.·34-s − 269.·35-s − 241.·37-s − 313.·38-s + 243.·40-s − 97.1·41-s + 376.·43-s + 179.·44-s + ⋯ |
L(s) = 1 | + 1.19·2-s + 0.425·4-s − 1.40·5-s + 0.926·7-s − 0.685·8-s − 1.67·10-s + 1.44·11-s + 1.10·14-s − 1.24·16-s + 1.01·17-s − 1.11·19-s − 0.598·20-s + 1.72·22-s − 1.72·23-s + 0.975·25-s + 0.394·28-s + 0.820·29-s − 0.0191·31-s − 0.800·32-s + 1.21·34-s − 1.30·35-s − 1.07·37-s − 1.33·38-s + 0.963·40-s − 0.370·41-s + 1.33·43-s + 0.616·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.049532426\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.049532426\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 3.37T + 8T^{2} \) |
| 5 | \( 1 + 15.7T + 125T^{2} \) |
| 7 | \( 1 - 17.1T + 343T^{2} \) |
| 11 | \( 1 - 52.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 3.29T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 97.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 349.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 903.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 131.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 556.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 254.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 183.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 780.T + 9.12e5T^{2} \) |
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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
−8.785773197578252747266804783170, −8.319109941153488334558639079977, −7.42457521291308871887368950710, −6.52021374438344614591871391848, −5.66313047363430451545399560697, −4.64841198676024256808005338840, −3.97382550066819854632361749135, −3.64164713111396805774403082064, −2.16705768273640376459538009434, −0.71439477097349914578675855941,
0.71439477097349914578675855941, 2.16705768273640376459538009434, 3.64164713111396805774403082064, 3.97382550066819854632361749135, 4.64841198676024256808005338840, 5.66313047363430451545399560697, 6.52021374438344614591871391848, 7.42457521291308871887368950710, 8.319109941153488334558639079977, 8.785773197578252747266804783170