| L(s) = 1 | + (−0.710 − 0.819i)3-s + (−9.64 + 1.38i)5-s + (−4.98 − 2.27i)7-s + (1.11 − 7.74i)9-s + (−3.69 + 12.5i)11-s + (0.537 + 1.17i)13-s + (7.98 + 6.92i)15-s + (3.26 − 5.07i)17-s + (−16.9 − 26.3i)19-s + (1.67 + 5.70i)21-s + (−15.3 + 17.1i)23-s + (67.1 − 19.7i)25-s + (−15.3 + 9.86i)27-s + (28.7 + 18.4i)29-s + (−5.00 + 5.77i)31-s + ⋯ |
| L(s) = 1 | + (−0.236 − 0.273i)3-s + (−1.92 + 0.277i)5-s + (−0.712 − 0.325i)7-s + (0.123 − 0.860i)9-s + (−0.335 + 1.14i)11-s + (0.0413 + 0.0904i)13-s + (0.532 + 0.461i)15-s + (0.191 − 0.298i)17-s + (−0.891 − 1.38i)19-s + (0.0798 + 0.271i)21-s + (−0.668 + 0.744i)23-s + (2.68 − 0.788i)25-s + (−0.568 + 0.365i)27-s + (0.992 + 0.637i)29-s + (−0.161 + 0.186i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00742472 - 0.137790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00742472 - 0.137790i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (15.3 - 17.1i)T \) |
| good | 3 | \( 1 + (0.710 + 0.819i)T + (-1.28 + 8.90i)T^{2} \) |
| 5 | \( 1 + (9.64 - 1.38i)T + (23.9 - 7.04i)T^{2} \) |
| 7 | \( 1 + (4.98 + 2.27i)T + (32.0 + 37.0i)T^{2} \) |
| 11 | \( 1 + (3.69 - 12.5i)T + (-101. - 65.4i)T^{2} \) |
| 13 | \( 1 + (-0.537 - 1.17i)T + (-110. + 127. i)T^{2} \) |
| 17 | \( 1 + (-3.26 + 5.07i)T + (-120. - 262. i)T^{2} \) |
| 19 | \( 1 + (16.9 + 26.3i)T + (-149. + 328. i)T^{2} \) |
| 29 | \( 1 + (-28.7 - 18.4i)T + (349. + 765. i)T^{2} \) |
| 31 | \( 1 + (5.00 - 5.77i)T + (-136. - 951. i)T^{2} \) |
| 37 | \( 1 + (21.1 + 3.04i)T + (1.31e3 + 385. i)T^{2} \) |
| 41 | \( 1 + (8.71 + 60.6i)T + (-1.61e3 + 473. i)T^{2} \) |
| 43 | \( 1 + (12.7 - 11.0i)T + (263. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + 40.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (60.6 + 27.7i)T + (1.83e3 + 2.12e3i)T^{2} \) |
| 59 | \( 1 + (-28.2 - 61.7i)T + (-2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (-2.76 - 2.39i)T + (529. + 3.68e3i)T^{2} \) |
| 67 | \( 1 + (-3.79 - 12.9i)T + (-3.77e3 + 2.42e3i)T^{2} \) |
| 71 | \( 1 + (43.2 - 12.6i)T + (4.24e3 - 2.72e3i)T^{2} \) |
| 73 | \( 1 + (51.5 - 33.1i)T + (2.21e3 - 4.84e3i)T^{2} \) |
| 79 | \( 1 + (-15.2 + 6.97i)T + (4.08e3 - 4.71e3i)T^{2} \) |
| 83 | \( 1 + (109. + 15.7i)T + (6.60e3 + 1.94e3i)T^{2} \) |
| 89 | \( 1 + (-38.5 + 33.4i)T + (1.12e3 - 7.84e3i)T^{2} \) |
| 97 | \( 1 + (-136. + 19.6i)T + (9.02e3 - 2.65e3i)T^{2} \) |
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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
−12.94752612253277187387218641604, −12.21113525669252919626428161976, −11.39915620277502462273724999151, −10.16019238908341379854915787140, −8.723050687855117312375261306379, −7.30863008659883182326476717478, −6.79303281973170662238978685694, −4.54428607871903005439988615836, −3.36041235013942729632076003621, −0.10880189022098034900550498248,
3.36520651176568323312002723551, 4.58173473880339902182298169479, 6.20879109038406234063184270662, 7.960997032920531001711088576792, 8.384611014057545742632950707003, 10.24361733244283712613590856301, 11.18227492865139953083743979603, 12.14714171751319268441508762492, 13.01797638743177754358498829579, 14.49133453904262737607035262382
