| L(s) = 1 | + 51.5i·5-s − 316. i·7-s + 455.·11-s − 2.32e3i·13-s − 5.96e3·17-s − 1.12e4·19-s + 7.52e3i·23-s + 1.29e4·25-s + 4.86e4i·29-s − 2.12e4i·31-s + 1.63e4·35-s + 1.28e4i·37-s − 1.92e4·41-s + 8.94e3·43-s − 1.62e5i·47-s + ⋯ |
| L(s) = 1 | + 0.412i·5-s − 0.923i·7-s + 0.342·11-s − 1.05i·13-s − 1.21·17-s − 1.64·19-s + 0.618i·23-s + 0.830·25-s + 1.99i·29-s − 0.712i·31-s + 0.380·35-s + 0.252i·37-s − 0.278·41-s + 0.112·43-s − 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.187178035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187178035\) |
| \(L(4)\) |
|
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 | \( 1 - 51.5iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 316. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 455.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.32e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.96e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.12e4T + 4.70e7T^{2} \) |
| 23 | \( 1 - 7.52e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.86e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.12e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.28e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.92e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 8.94e3T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.62e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.67e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.36e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.54e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.91e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.72e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.86e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.47e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 9.26e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 8.21e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 6.75e5T + 8.32e11T^{2} \) |
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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
−10.22458330815974400151820617882, −8.975586432095981985504924894644, −8.237829045978290583986941905119, −7.06172887614931866857251664471, −6.61814939798807067940142314482, −5.33368381832445639422189181952, −4.25377214444778399546182533812, −3.36285917473437055635502606518, −2.14587085117259067422770930697, −0.841376478250106028102264555001,
0.28430759593207438445053091613, 1.81058394914929413336879281688, 2.56792069968881812969527927880, 4.18111105796711574373096532352, 4.75213002079776567937824722970, 6.17212752317948892947177798012, 6.59722836783375587135527063379, 8.005482281491725582436987475729, 8.948160052719275206418545285474, 9.188303709763479350524806709324
