| L(s) = 1 | + 6i·5-s + 21.1·7-s + 42.3i·11-s + 20i·13-s + 8·17-s + 84.6i·19-s − 169.·23-s + 89·25-s + 46i·29-s − 21.1·31-s + 126. i·35-s + 164i·37-s − 312·41-s − 423. i·43-s + 169.·47-s + ⋯ |
| L(s) = 1 | + 0.536i·5-s + 1.14·7-s + 1.16i·11-s + 0.426i·13-s + 0.114·17-s + 1.02i·19-s − 1.53·23-s + 0.711·25-s + 0.294i·29-s − 0.122·31-s + 0.613i·35-s + 0.728i·37-s − 1.18·41-s − 1.50i·43-s + 0.525·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.677747977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677747977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6iT - 125T^{2} \) |
| 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 - 42.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 21.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 164iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 312T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 266iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 253. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 132iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 507. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 677.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 246T + 3.89e5T^{2} \) |
| 79 | \( 1 - 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 973. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 302T + 9.12e5T^{2} \) |
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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
−9.967387668039059270002992455457, −8.810039742906898247669518939500, −8.020924988726755049364244647574, −7.31151455430465009712129280574, −6.49269611964710388689063653938, −5.40490202981920661751287285999, −4.55380884912313213330433087619, −3.67275405902544004804967415232, −2.24457335923841451895262156729, −1.51086005701955343445194400258,
0.39322742959378937911642672746, 1.45568381472207773888265977932, 2.70417249903300656061131754117, 3.91959040054945740303870633145, 4.88867339007516906535949521292, 5.56590742878226441953638798846, 6.53481314099422948401845256461, 7.73413688375966024531533331511, 8.298832415162704569261921318496, 8.917826352644620618213175690143
