| L(s) = 1 | + 5.65·2-s + (−16.9 + 21i)3-s + 32.0·4-s + (−96 + 118. i)6-s − 2i·7-s + 181.·8-s + (−153. − 712. i)9-s − 33.9i·11-s + (−543. + 672. i)12-s − 2.95e3i·13-s − 11.3i·14-s + 1.02e3·16-s + 4.48e3·17-s + (−865. − 4.03e3i)18-s − 5.25e3·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.628 + 0.777i)3-s + 0.500·4-s + (−0.444 + 0.549i)6-s − 0.00583i·7-s + 0.353·8-s + (−0.209 − 0.977i)9-s − 0.0255i·11-s + (−0.314 + 0.388i)12-s − 1.34i·13-s − 0.00412i·14-s + 0.250·16-s + 0.911·17-s + (−0.148 − 0.691i)18-s − 0.766·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.434331213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434331213\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 + (16.9 - 21i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 33.9iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.95e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.48e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 5.25e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.02e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.20e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.28e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.40e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.67e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.40e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.79e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 1.92e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 3.26e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 6.25e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.38e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 6.82e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.30e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 4.96e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 3.86e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.81e5iT - 8.32e11T^{2} \) |
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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
−11.98050608429027112561378199868, −10.76255960472433456355038051181, −10.24304238184847131670793480121, −8.850836380444761539965402845837, −7.43128589632575662611403432942, −6.03572791941070882267950991620, −5.27275761593684370852909798591, −4.06935168054215949448167112089, −2.89364106199141203766927317079, −0.73103366940431477691553198266,
1.15234416797797178852660397689, 2.48386441672522719088108839671, 4.22870332500872164175966606879, 5.42191476091708382183590940833, 6.50878272372524157608700327732, 7.31292530974798972108424608486, 8.633797639509883731084068109088, 10.19174063170269781201035091898, 11.30133218451464083183719751144, 12.00258433799300033904321719251
