| L(s) = 1 | + (−4.86 + 6.35i)2-s + 15.5·3-s + (−16.6 − 61.7i)4-s − 100. i·5-s + (−75.8 + 98.9i)6-s − 277. i·7-s + (473. + 194. i)8-s + 243·9-s + (640. + 490. i)10-s + 1.92e3·11-s + (−259. − 963. i)12-s − 1.72e3i·13-s + (1.76e3 + 1.35e3i)14-s − 1.57e3i·15-s + (−3.54e3 + 2.05e3i)16-s + 1.65e3·17-s + ⋯ |
| L(s) = 1 | + (−0.608 + 0.793i)2-s + 0.577·3-s + (−0.260 − 0.965i)4-s − 0.806i·5-s + (−0.351 + 0.458i)6-s − 0.809i·7-s + (0.924 + 0.380i)8-s + 0.333·9-s + (0.640 + 0.490i)10-s + 1.44·11-s + (−0.150 − 0.557i)12-s − 0.783i·13-s + (0.642 + 0.492i)14-s − 0.465i·15-s + (−0.864 + 0.502i)16-s + 0.336·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.32807 - 0.262529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32807 - 0.262529i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.86 - 6.35i)T \) |
| 3 | \( 1 - 15.5T \) |
| good | 5 | \( 1 + 100. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 277. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.92e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.72e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 1.65e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.36e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.56e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.84e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.02e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.25e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 3.41e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 7.12e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.49e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.35e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.35e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 1.06e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.07e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.45e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.35e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.60e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 9.42e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.16e6T + 8.32e11T^{2} \) |
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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
−16.64818799513144551939656777510, −15.07307939319641936411959035555, −14.12264528836642146586355736826, −12.74097563226540873795412528128, −10.58812313914034134597800396458, −9.187184588457061369450001384055, −8.139954074411440933030519777551, −6.56133399882507671285505979024, −4.43912054354703705162433216843, −1.04885376408749071552740768687,
2.03302768855523391978008002563, 3.74378858056220502476601432111, 6.84107863621455815823785453193, 8.610407732253279484490301591344, 9.639807202000321828934098650299, 11.20575881272298960254425773609, 12.30611919924025232951976519510, 13.93288808807711946033222484938, 15.07847938603603811631602708226, 16.74250348389591648092254515211
