| L(s) = 1 | + 12i·5-s − 5.19i·7-s − 62.3·11-s + 7·13-s − 84i·17-s − 67.5i·19-s + 62.3·23-s − 19·25-s + 168i·29-s − 259. i·31-s + 62.3·35-s − 97·37-s + 72i·41-s − 363. i·43-s + 436.·47-s + ⋯ |
| L(s) = 1 | + 1.07i·5-s − 0.280i·7-s − 1.70·11-s + 0.149·13-s − 1.19i·17-s − 0.815i·19-s + 0.565·23-s − 0.151·25-s + 1.07i·29-s − 1.50i·31-s + 0.301·35-s − 0.430·37-s + 0.274i·41-s − 1.28i·43-s + 1.35·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9553320739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9553320739\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 12iT - 125T^{2} \) |
| 7 | \( 1 + 5.19iT - 343T^{2} \) |
| 11 | \( 1 + 62.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 67.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 62.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 168iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 259. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 97T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 363. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 504iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 436.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 133T + 2.26e5T^{2} \) |
| 67 | \( 1 + 545. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 498.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 497T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 872.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 749T + 9.12e5T^{2} \) |
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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
−10.71610495085320048515307039527, −9.775855710213901088401174956185, −8.681656755011875659102563183095, −7.41704541465141603347359380858, −7.07735965537212970131979886703, −5.70919717186085496814726644053, −4.74030107688681728204541616298, −3.19773614654381540334278990412, −2.43437737192559299746085772776, −0.32010863784303315302344114905,
1.30808233342821433754334649053, 2.75428789836025601458375383294, 4.23720208086741363136866149753, 5.25681093951666792916615351056, 5.98236897263922781325220516598, 7.49622239077110795197593902357, 8.325333440291169467419816324347, 8.951704090957151616435704693036, 10.19543368508928378180129719153, 10.76338573240326293168378050100
