| L(s) = 1 | + 4.42·5-s − 3·9-s + 3.52i·11-s + 5.32i·19-s + 4.79i·23-s + 14.5·25-s + 9.59i·31-s + 2.61·37-s + 9.59·41-s − 12.3i·43-s − 13.2·45-s − 2i·47-s − 7·49-s + 6.23·53-s + 15.5i·55-s + ⋯ |
| L(s) = 1 | + 1.97·5-s − 9-s + 1.06i·11-s + 1.22i·19-s + 0.999i·23-s + 2.91·25-s + 1.72i·31-s + 0.430·37-s + 1.49·41-s − 1.88i·43-s − 1.97·45-s − 0.291i·47-s − 49-s + 0.856·53-s + 2.10i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.222240644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.222240644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 4.79iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 4.42T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.52iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.32iT - 19T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.59iT - 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 9.59T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.465918709062592385945592322710, −9.139133039623096790268235680172, −8.083326524300774935540889059246, −7.02146780332575562938675769936, −6.17732647364271088286129971948, −5.55332848286498700444883863053, −4.88847454799900023949314353629, −3.38226256236228159172567390728, −2.29499094838776145664633418840, −1.53735942489445607249577982380,
0.909859837434041623540232811268, 2.45119549870087173472430686598, 2.85076866599662300525967976877, 4.51875124217609813832209791576, 5.53590520485991296547326191110, 6.04401796387356927585931221226, 6.61215256741919537064627512584, 7.961519389871490892053279642664, 8.860431741138508490522613762010, 9.320352945145423844753894621908
