| L(s) = 1 | + (−2.54 − 4.52i)3-s + (−3.81 − 6.60i)5-s + (−15.6 − 9.84i)7-s + (−14.0 + 23.0i)9-s + (−28.6 − 16.5i)11-s + (31.2 + 18.0i)13-s + (−20.2 + 34.1i)15-s + (24.8 + 43.0i)17-s + (−16.5 − 9.54i)19-s + (−4.64 + 96.1i)21-s + (−35.5 + 20.5i)23-s + (33.3 − 57.8i)25-s + (140. + 4.74i)27-s + (70.5 − 40.7i)29-s + 213. i·31-s + ⋯ |
| L(s) = 1 | + (−0.490 − 0.871i)3-s + (−0.341 − 0.591i)5-s + (−0.846 − 0.531i)7-s + (−0.519 + 0.854i)9-s + (−0.786 − 0.454i)11-s + (0.667 + 0.385i)13-s + (−0.347 + 0.587i)15-s + (0.354 + 0.614i)17-s + (−0.199 − 0.115i)19-s + (−0.0482 + 0.998i)21-s + (−0.322 + 0.186i)23-s + (0.267 − 0.462i)25-s + (0.999 + 0.0338i)27-s + (0.452 − 0.260i)29-s + 1.23i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0118 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0118 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1825122764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1825122764\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.54 + 4.52i)T \) |
| 7 | \( 1 + (15.6 + 9.84i)T \) |
| good | 5 | \( 1 + (3.81 + 6.60i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (28.6 + 16.5i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-31.2 - 18.0i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-24.8 - 43.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (16.5 + 9.54i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (35.5 - 20.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-70.5 + 40.7i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 213. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (94.2 - 163. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (101. - 176. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-116. - 202. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 291.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-3.42 + 1.97i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 557.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 185. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 522.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 876. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (765. - 441. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 816.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (276. + 478. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (225. - 390. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (337. - 194. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.00355530578111463011475654575, −10.94889156345680157320842812048, −10.12605617702863658113788036766, −8.650166333655797407778258642001, −7.937020039993331932801901790567, −6.75107145669053050199495764958, −5.94042756460201123319031663863, −4.63490110679666449992936324829, −3.11785247671613178615990273188, −1.25926598785408439955988785800,
0.083172323934658210615349821256, 2.79282450809127878103568010344, 3.80205629410325611746926180851, 5.22671106697621753517239076827, 6.13416060785993852027125772792, 7.27261321674967359025996932712, 8.612573776937664069171829732288, 9.662354678736312325921752936547, 10.40174061778495987291536529614, 11.22695376214962158863779015079
