| L(s) = 1 | + 1.36i·2-s + 3-s + 0.128·4-s − 2.18i·5-s + 1.36i·6-s − 4.27i·7-s + 2.91i·8-s + 9-s + 2.98·10-s + i·11-s + 0.128·12-s + (−0.922 − 3.48i)13-s + 5.84·14-s − 2.18i·15-s − 3.72·16-s + 3.79·17-s + ⋯ |
| L(s) = 1 | + 0.967i·2-s + 0.577·3-s + 0.0640·4-s − 0.976i·5-s + 0.558i·6-s − 1.61i·7-s + 1.02i·8-s + 0.333·9-s + 0.944·10-s + 0.301i·11-s + 0.0370·12-s + (−0.255 − 0.966i)13-s + 1.56·14-s − 0.563i·15-s − 0.931·16-s + 0.919·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85525 + 0.241341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85525 + 0.241341i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (0.922 + 3.48i)T \) |
| good | 2 | \( 1 - 1.36iT - 2T^{2} \) |
| 5 | \( 1 + 2.18iT - 5T^{2} \) |
| 7 | \( 1 + 4.27iT - 7T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 - 3.17iT - 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 + 0.0727iT - 31T^{2} \) |
| 37 | \( 1 - 3.66iT - 37T^{2} \) |
| 41 | \( 1 - 4.16iT - 41T^{2} \) |
| 43 | \( 1 + 7.11T + 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 - 5.62iT - 59T^{2} \) |
| 61 | \( 1 + 5.40T + 61T^{2} \) |
| 67 | \( 1 + 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 9.62iT - 71T^{2} \) |
| 73 | \( 1 - 6.44iT - 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 7.80iT - 83T^{2} \) |
| 89 | \( 1 - 2.87iT - 89T^{2} \) |
| 97 | \( 1 + 3.79iT - 97T^{2} \) |
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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
−11.02724820522840545272226348797, −10.19011643836144932740455857594, −9.223776444078252151474443324699, −7.976920086131455179159461176841, −7.72706396401348546409622358645, −6.76281216606276824147198774015, −5.46235661108231291887080850284, −4.56522898148011419524865355262, −3.26024751275536021365570087787, −1.30153165306499893560337598840,
1.98628983682244623158151723082, 2.77427391614251054057964748137, 3.61943689996027661652359597756, 5.32209884418142877255568229670, 6.57755472582170690130764530938, 7.33413091935819557464902887654, 8.783960945478763714355198237363, 9.366321098842332248136258447738, 10.32477811306648995158889759658, 11.28819782197258415367476720139
