| L(s) = 1 | + 2·5-s + 9.79i·7-s + 19.5i·11-s − 21·25-s − 50·29-s + 48.9i·31-s + 19.5i·35-s − 46.9·49-s + 94·53-s + 39.1i·55-s − 117. i·59-s − 50·73-s − 191.·77-s − 146. i·79-s − 97.9i·83-s + ⋯ |
| L(s) = 1 | + 0.400·5-s + 1.39i·7-s + 1.78i·11-s − 0.839·25-s − 1.72·29-s + 1.58i·31-s + 0.559i·35-s − 0.959·49-s + 1.77·53-s + 0.712i·55-s − 1.99i·59-s − 0.684·73-s − 2.49·77-s − 1.86i·79-s − 1.18i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.113758883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113758883\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2T + 25T^{2} \) |
| 7 | \( 1 - 9.79iT - 49T^{2} \) |
| 11 | \( 1 - 19.5iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 50T + 841T^{2} \) |
| 31 | \( 1 - 48.9iT - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 94T + 2.80e3T^{2} \) |
| 59 | \( 1 + 117. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 146. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 97.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−9.252323473804554270994662583211, −8.603258325580597282536607305726, −7.59664215659282173926497508411, −6.93984244779167864714567609113, −5.97133742641685569718494832400, −5.32174954055330780145144605827, −4.57252794877238667427058945948, −3.41425095394894885275083059965, −2.21548733895160069777541257204, −1.78068829062375822885783506935,
0.26688735975999893160163739402, 1.19926428104994108417954238602, 2.52211840332669157915131240029, 3.72107313182437970526660117334, 4.09818071117764543335843482563, 5.54040420567272753428225089478, 5.92937407018795759817014023577, 6.97379956533045215473993456100, 7.65795639840418766304199506872, 8.379484714569256146536086506080
