| L(s) = 1 | + (3.74 + 1.80i)2-s + (−13.9 + 17.5i)3-s + (−9.16 − 11.4i)4-s + (−40.2 − 19.3i)5-s + (−83.8 + 40.4i)6-s + (−30.7 + 38.5i)7-s + (−43.2 − 189. i)8-s + (−57.4 − 251. i)9-s + (−115. − 145. i)10-s + (−124. + 546. i)11-s + 328.·12-s + (−134. + 587. i)13-s + (−184. + 88.8i)14-s + (900. − 433. i)15-s + (75.1 − 329. i)16-s − 1.32e3·17-s + ⋯ |
| L(s) = 1 | + (0.662 + 0.319i)2-s + (−0.895 + 1.12i)3-s + (−0.286 − 0.359i)4-s + (−0.720 − 0.346i)5-s + (−0.951 + 0.458i)6-s + (−0.236 + 0.296i)7-s + (−0.238 − 1.04i)8-s + (−0.236 − 1.03i)9-s + (−0.366 − 0.459i)10-s + (−0.310 + 1.36i)11-s + 0.659·12-s + (−0.220 + 0.963i)13-s + (−0.251 + 0.121i)14-s + (1.03 − 0.497i)15-s + (0.0734 − 0.321i)16-s − 1.11·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.148i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0351252 - 0.469257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0351252 - 0.469257i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-4.18e3 - 1.74e3i)T \) |
| good | 2 | \( 1 + (-3.74 - 1.80i)T + (19.9 + 25.0i)T^{2} \) |
| 3 | \( 1 + (13.9 - 17.5i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (40.2 + 19.3i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + (30.7 - 38.5i)T + (-3.73e3 - 1.63e4i)T^{2} \) |
| 11 | \( 1 + (124. - 546. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (134. - 587. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-375. - 470. i)T + (-5.50e5 + 2.41e6i)T^{2} \) |
| 23 | \( 1 + (-3.63e3 + 1.74e3i)T + (4.01e6 - 5.03e6i)T^{2} \) |
| 31 | \( 1 + (2.25e3 + 1.08e3i)T + (1.78e7 + 2.23e7i)T^{2} \) |
| 37 | \( 1 + (217. + 952. i)T + (-6.24e7 + 3.00e7i)T^{2} \) |
| 41 | \( 1 + 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (8.70e3 - 4.18e3i)T + (9.16e7 - 1.14e8i)T^{2} \) |
| 47 | \( 1 + (2.26e3 - 9.94e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.20e4 + 5.82e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 - 3.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-1.49e3 + 1.87e3i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.53e4 - 6.74e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-1.67e4 + 7.35e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (6.10e4 - 2.94e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + (2.47e3 + 1.08e4i)T + (-2.77e9 + 1.33e9i)T^{2} \) |
| 83 | \( 1 + (5.04e4 + 6.33e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-2.98e4 - 1.43e4i)T + (3.48e9 + 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-7.79e4 - 9.77e4i)T + (-1.91e9 + 8.37e9i)T^{2} \) |
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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
−16.24900822181099514666344068541, −15.55908524995486533028195348389, −14.68353597838677462742472631296, −12.91225711884741016305766116424, −11.75072667612399551356159656974, −10.30410491430087509451851517046, −9.174063415719382279483610193522, −6.70046252130915082775850507658, −4.96344909496645114293625313264, −4.31637032263142261977782609405,
0.26828926521359179449403380323, 3.25309568855493057991067936892, 5.36363379118506566025731517999, 7.00962535648863603986652031734, 8.347995097552663094340839848350, 11.02627737817193605002983724332, 11.72211880377913141737692775869, 13.01351136885553101721753410917, 13.56270999406734033525485797446, 15.37447098316660166237758416388
