| L(s) = 1 | + (−3.10 + 0.910i)2-s + (−3.16 − 3.65i)3-s + (5.42 − 3.48i)4-s + (−5.40 + 0.777i)5-s + (13.1 + 8.45i)6-s + (−0.889 − 0.406i)7-s + (−5.17 + 5.97i)8-s + (−2.05 + 14.2i)9-s + (16.0 − 7.33i)10-s + (3.66 − 12.4i)11-s + (−29.9 − 8.79i)12-s + (−5.85 − 12.8i)13-s + (3.12 + 0.449i)14-s + (19.9 + 17.3i)15-s + (−0.0933 + 0.204i)16-s + (−7.74 + 12.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.55 + 0.455i)2-s + (−1.05 − 1.21i)3-s + (1.35 − 0.871i)4-s + (−1.08 + 0.155i)5-s + (2.19 + 1.40i)6-s + (−0.127 − 0.0580i)7-s + (−0.647 + 0.747i)8-s + (−0.228 + 1.58i)9-s + (1.60 − 0.733i)10-s + (0.333 − 1.13i)11-s + (−2.49 − 0.732i)12-s + (−0.450 − 0.986i)13-s + (0.223 + 0.0321i)14-s + (1.33 + 1.15i)15-s + (−0.00583 + 0.0127i)16-s + (−0.455 + 0.708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0643604 - 0.162704i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0643604 - 0.162704i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-14.0 + 18.2i)T \) |
| good | 2 | \( 1 + (3.10 - 0.910i)T + (3.36 - 2.16i)T^{2} \) |
| 3 | \( 1 + (3.16 + 3.65i)T + (-1.28 + 8.90i)T^{2} \) |
| 5 | \( 1 + (5.40 - 0.777i)T + (23.9 - 7.04i)T^{2} \) |
| 7 | \( 1 + (0.889 + 0.406i)T + (32.0 + 37.0i)T^{2} \) |
| 11 | \( 1 + (-3.66 + 12.4i)T + (-101. - 65.4i)T^{2} \) |
| 13 | \( 1 + (5.85 + 12.8i)T + (-110. + 127. i)T^{2} \) |
| 17 | \( 1 + (7.74 - 12.0i)T + (-120. - 262. i)T^{2} \) |
| 19 | \( 1 + (3.48 + 5.42i)T + (-149. + 328. i)T^{2} \) |
| 29 | \( 1 + (2.47 + 1.59i)T + (349. + 765. i)T^{2} \) |
| 31 | \( 1 + (-5.65 + 6.52i)T + (-136. - 951. i)T^{2} \) |
| 37 | \( 1 + (-30.2 - 4.34i)T + (1.31e3 + 385. i)T^{2} \) |
| 41 | \( 1 + (4.02 + 28.0i)T + (-1.61e3 + 473. i)T^{2} \) |
| 43 | \( 1 + (25.3 - 21.9i)T + (263. - 1.83e3i)T^{2} \) |
| 47 | \( 1 - 28.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-74.4 - 34.0i)T + (1.83e3 + 2.12e3i)T^{2} \) |
| 59 | \( 1 + (37.0 + 81.0i)T + (-2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (-13.9 - 12.1i)T + (529. + 3.68e3i)T^{2} \) |
| 67 | \( 1 + (23.0 + 78.6i)T + (-3.77e3 + 2.42e3i)T^{2} \) |
| 71 | \( 1 + (70.4 - 20.6i)T + (4.24e3 - 2.72e3i)T^{2} \) |
| 73 | \( 1 + (62.8 - 40.3i)T + (2.21e3 - 4.84e3i)T^{2} \) |
| 79 | \( 1 + (-109. + 50.0i)T + (4.08e3 - 4.71e3i)T^{2} \) |
| 83 | \( 1 + (119. + 17.1i)T + (6.60e3 + 1.94e3i)T^{2} \) |
| 89 | \( 1 + (25.2 - 21.8i)T + (1.12e3 - 7.84e3i)T^{2} \) |
| 97 | \( 1 + (-118. + 17.0i)T + (9.02e3 - 2.65e3i)T^{2} \) |
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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
−17.30433969119953333679284270959, −16.54326085171873149969862359803, −15.27064174337173700798633275750, −12.98869409551319436661886696253, −11.57472239162345238849231158693, −10.66763363448340068793369856176, −8.449316190106097608996931969887, −7.42269193232644735701493184537, −6.23326970348666292493180466705, −0.45267105843507414576271785190,
4.48070990125311463511153044842, 7.22495480804785723411634760758, 9.120880159364381018780673619724, 10.05892074491562622005268897581, 11.39752489375512852820221949224, 11.94224279687742977524073657032, 15.14729157596384437642096741802, 16.17005192345476971730183622883, 16.92550506495372612356960625651, 17.93882915794141724215502853143
