| L(s) = 1 | + 2i·5-s + 9.79i·7-s + 19.5·11-s + 21·25-s + 50i·29-s − 48.9i·31-s − 19.5·35-s − 46.9·49-s + 94i·53-s + 39.1i·55-s − 117.·59-s + 50·73-s + 191. i·77-s + 146. i·79-s + 97.9·83-s + ⋯ |
| L(s) = 1 | + 0.400i·5-s + 1.39i·7-s + 1.78·11-s + 0.839·25-s + 1.72i·29-s − 1.58i·31-s − 0.559·35-s − 0.959·49-s + 1.77i·53-s + 0.712i·55-s − 1.99·59-s + 0.684·73-s + 2.49i·77-s + 1.86i·79-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.034937300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034937300\) |
| \(L(2)\) |
|
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 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 - 9.79iT - 49T^{2} \) |
| 11 | \( 1 - 19.5T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 50iT - 841T^{2} \) |
| 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 | \( 1 - 94iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 117.T + 3.48e3T^{2} \) |
| 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 | \( 1 - 97.9T + 6.88e3T^{2} \) |
| 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
−9.466309394877752989834197825896, −9.146015386788151476522327629068, −8.357355243561090251645568465293, −7.19811123482758943996720094233, −6.40256164978630995380455819596, −5.74194017524548546685894363839, −4.64071195967178653338179789815, −3.54006879705857753769415358668, −2.54688546269181576070764416388, −1.35677478127816702379097429599,
0.68079489459746825856764101310, 1.61123040821872616575888505994, 3.35846041890088462009654509504, 4.14826242169324120249194317948, 4.87615764422648190118350236222, 6.27504876914897286755346824105, 6.83483314747049142690532555878, 7.70562536347938761484195003161, 8.660070369244321336374511831787, 9.398830896493685616318733553266
