| L(s) = 1 | + 2.17·2-s − 3.26·4-s − 7.52·5-s − 1.76·7-s − 24.5·8-s − 16.3·10-s − 0.114·11-s − 11.7·13-s − 3.84·14-s − 27.2·16-s − 131.·17-s − 42.0·19-s + 24.5·20-s − 0.249·22-s − 84.1·23-s − 68.3·25-s − 25.5·26-s + 5.76·28-s + 0.308·29-s + 143.·31-s + 136.·32-s − 286.·34-s + 13.3·35-s − 161.·37-s − 91.4·38-s + 184.·40-s + 136.·41-s + ⋯ |
| L(s) = 1 | + 0.769·2-s − 0.407·4-s − 0.673·5-s − 0.0954·7-s − 1.08·8-s − 0.518·10-s − 0.00314·11-s − 0.250·13-s − 0.0734·14-s − 0.425·16-s − 1.87·17-s − 0.507·19-s + 0.274·20-s − 0.00242·22-s − 0.762·23-s − 0.546·25-s − 0.192·26-s + 0.0389·28-s + 0.00197·29-s + 0.831·31-s + 0.755·32-s − 1.44·34-s + 0.0642·35-s − 0.716·37-s − 0.390·38-s + 0.729·40-s + 0.518·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7767054218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7767054218\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 2.17T + 8T^{2} \) |
| 5 | \( 1 + 7.52T + 125T^{2} \) |
| 7 | \( 1 + 1.76T + 343T^{2} \) |
| 11 | \( 1 + 0.114T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 0.308T + 2.43e4T^{2} \) |
| 31 | \( 1 - 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 71.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 412.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 135.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 621.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 616.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 493.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 321.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 400.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 387.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 702.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 994.T + 9.12e5T^{2} \) |
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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
−8.613214238621659489055447137532, −8.109449442014215281670243297065, −6.97565455184505654065420719230, −6.31868678808441704514308506676, −5.42045804871965974440991191923, −4.41009672953278647198227878219, −4.14100814210623313803055924345, −3.07945589299985622544070957898, −2.06037177869924372825511056798, −0.33716582564535775845871757838,
0.33716582564535775845871757838, 2.06037177869924372825511056798, 3.07945589299985622544070957898, 4.14100814210623313803055924345, 4.41009672953278647198227878219, 5.42045804871965974440991191923, 6.31868678808441704514308506676, 6.97565455184505654065420719230, 8.109449442014215281670243297065, 8.613214238621659489055447137532
