L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−4.69 − 8.12i)5-s − 7.99·8-s + (9.38 − 16.2i)10-s + (10 − 17.3i)11-s − 65.6·13-s + (−8 − 13.8i)16-s + (−28.1 + 48.7i)17-s + (4.69 + 8.12i)19-s + 37.5·20-s + 40·22-s + (24 + 41.5i)23-s + (18.4 − 32.0i)25-s + (−65.6 − 113. i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.419 − 0.726i)5-s − 0.353·8-s + (0.296 − 0.513i)10-s + (0.274 − 0.474i)11-s − 1.40·13-s + (−0.125 − 0.216i)16-s + (−0.401 + 0.695i)17-s + (0.0566 + 0.0980i)19-s + 0.419·20-s + 0.387·22-s + (0.217 + 0.376i)23-s + (0.147 − 0.256i)25-s + (−0.495 − 0.857i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.752481860\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752481860\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (4.69 + 8.12i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-10 + 17.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (28.1 - 48.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-4.69 - 8.12i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-24 - 41.5i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + (103. - 178. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-39 - 67.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436T + 7.95e4T^{2} \) |
| 47 | \( 1 + (103. + 178. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-31 + 53.6i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-333. + 576. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-136. - 235. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (290 - 502. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 544T + 3.57e5T^{2} \) |
| 73 | \( 1 + (300. - 519. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-340 - 588. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-750. - 1.29e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 656.T + 9.12e5T^{2} \) |
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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
−9.717606823518858218787747835032, −8.857871145351734312584870796024, −8.199773988763092860892488159844, −7.34141048171107495229684780436, −6.48403248122904204447226352379, −5.42861670696303122012108369333, −4.66515685763243287263335763480, −3.81690643665550885140924116491, −2.51706869632508989809956534808, −0.830220883700939078431256675228,
0.55267434754114883875899763277, 2.24614074098158462520568644211, 2.94334116871317868461820499553, 4.19094983641107607890686399542, 4.88787473516630609521985663258, 6.08685302368094184914655572498, 7.13891062815794899594573307045, 7.64842535925829597575679915052, 9.085133722152343393647118143908, 9.646968431969173466397210473914