| L(s) = 1 | − 1.99·2-s + 10.2·3-s − 4.03·4-s − 6.84·5-s − 20.3·6-s − 3.06·7-s + 23.9·8-s + 77.1·9-s + 13.6·10-s − 55.5·11-s − 41.1·12-s + 26.7·13-s + 6.11·14-s − 69.8·15-s − 15.4·16-s + 40.1·17-s − 153.·18-s + 64.0·19-s + 27.5·20-s − 31.3·21-s + 110.·22-s − 111.·23-s + 244.·24-s − 78.1·25-s − 53.2·26-s + 512.·27-s + 12.3·28-s + ⋯ |
| L(s) = 1 | − 0.704·2-s + 1.96·3-s − 0.504·4-s − 0.612·5-s − 1.38·6-s − 0.165·7-s + 1.05·8-s + 2.85·9-s + 0.431·10-s − 1.52·11-s − 0.990·12-s + 0.570·13-s + 0.116·14-s − 1.20·15-s − 0.241·16-s + 0.572·17-s − 2.01·18-s + 0.773·19-s + 0.308·20-s − 0.325·21-s + 1.07·22-s − 1.00·23-s + 2.08·24-s − 0.625·25-s − 0.401·26-s + 3.65·27-s + 0.0834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.324001250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.324001250\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.99T + 8T^{2} \) |
| 3 | \( 1 - 10.2T + 27T^{2} \) |
| 5 | \( 1 + 6.84T + 125T^{2} \) |
| 7 | \( 1 + 3.06T + 343T^{2} \) |
| 11 | \( 1 + 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 126.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 46.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 376.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 612.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 292.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 270.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 340.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 56.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 746.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 493.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 233.T + 9.12e5T^{2} \) |
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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
−8.712370476566998960680285316139, −8.131844776253104090834417729028, −7.74639664010668600705661764461, −7.19518782266594743337390218795, −5.56752231050426779959979758004, −4.39824599735702838207966994204, −3.76348766918615697324679665828, −2.91556893848381811657696397359, −1.92959475424059522754388205579, −0.73188238373689531154942414321,
0.73188238373689531154942414321, 1.92959475424059522754388205579, 2.91556893848381811657696397359, 3.76348766918615697324679665828, 4.39824599735702838207966994204, 5.56752231050426779959979758004, 7.19518782266594743337390218795, 7.74639664010668600705661764461, 8.131844776253104090834417729028, 8.712370476566998960680285316139
