| L(s) = 1 | + 4.54i·5-s + 61.7i·7-s + 130.·11-s − 136. i·13-s + 298.·17-s − 301.·19-s + 471. i·23-s + 604.·25-s + 648. i·29-s − 693. i·31-s − 280.·35-s − 1.50e3i·37-s + 2.30e3·41-s + 985.·43-s − 1.76e3i·47-s + ⋯ |
| L(s) = 1 | + 0.181i·5-s + 1.25i·7-s + 1.07·11-s − 0.810i·13-s + 1.03·17-s − 0.835·19-s + 0.892i·23-s + 0.966·25-s + 0.771i·29-s − 0.721i·31-s − 0.229·35-s − 1.09i·37-s + 1.37·41-s + 0.532·43-s − 0.799i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.527792543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527792543\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.54iT - 625T^{2} \) |
| 7 | \( 1 - 61.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 130.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 136. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 298.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 301.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 471. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 648. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 693. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.50e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.30e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 985.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.76e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 717. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.92e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 6.48e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.37e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.07e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.68e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.22e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.43e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 3.14e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.21e4T + 8.85e7T^{2} \) |
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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
−9.240765456921262654071404182440, −8.660517971424396698344201330867, −7.75571473078387392306838908765, −6.80668408927409617179394846540, −5.84370803837791043146937768616, −5.33943585737910263527775365400, −4.02200663565386242578696443703, −3.08264708048405333812986445650, −2.08626196706101029425305575409, −0.863323227264423437561303321789,
0.70674016663498596910833400041, 1.48044084372056896676082560752, 2.93686305498994643320530846847, 4.17933661265525689047642583901, 4.44882886195193885899088352880, 5.91547830134835720818425609738, 6.73959068412667166654631970951, 7.35384221500816146005640896098, 8.392256010081239704873361422444, 9.115695585042138540361466275166
