| L(s) = 1 | + (−1.13 − 5.06i)3-s − 14.6i·5-s − 7i·7-s + (−24.4 + 11.5i)9-s − 13.9·11-s − 45.4·13-s + (−74.2 + 16.6i)15-s − 120. i·17-s + 34.9i·19-s + (−35.4 + 7.97i)21-s + 151.·23-s − 89.4·25-s + (86.3 + 110. i)27-s + 224. i·29-s + 44.2i·31-s + ⋯ |
| L(s) = 1 | + (−0.219 − 0.975i)3-s − 1.30i·5-s − 0.377i·7-s + (−0.903 + 0.427i)9-s − 0.381·11-s − 0.970·13-s + (−1.27 + 0.287i)15-s − 1.72i·17-s + 0.421i·19-s + (−0.368 + 0.0829i)21-s + 1.37·23-s − 0.715·25-s + (0.615 + 0.787i)27-s + 1.43i·29-s + 0.256i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6997015636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6997015636\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.13 + 5.06i)T \) |
| 7 | \( 1 + 7iT \) |
| good | 5 | \( 1 + 14.6iT - 125T^{2} \) |
| 11 | \( 1 + 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 34.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 44.2iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 459. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 497. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 282. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 48.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 343.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 678. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 820.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 370.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 986. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 484.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 980. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 488.T + 9.12e5T^{2} \) |
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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
−10.67054670325773571655231064770, −9.381426580121766260545435810354, −8.643291909570550875996594450945, −7.53133798964439626063703561136, −6.91093310398367522937998956565, −5.29936108270024948423487162888, −4.88422619295626616265344708190, −2.91360226460320984325529910288, −1.37948292368763331979021040670, −0.25997401602376353881873606838,
2.46248243205416119986794065395, 3.43624138732661634270978255648, 4.70860888293418112641603760312, 5.85408200483469441658808572322, 6.76682609962027981104555248554, 7.969294152537124220860480951060, 9.103968997876438324416334508959, 10.10832955613904841784893285153, 10.66347143418641812340902645630, 11.42270845583083110519118282663
