L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s − 3i·7-s − i·8-s + 9-s + 10-s + 0.267i·11-s + 12-s + 3·14-s + i·15-s + 16-s + 4·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 1.13i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.0807i·11-s + 0.288·12-s + 0.801·14-s + 0.258i·15-s + 0.250·16-s + 0.970·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.288406290\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.288406290\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 0.267iT - 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 5.73iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 4.92iT - 31T^{2} \) |
| 37 | \( 1 + 5.92iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 6.46iT - 47T^{2} \) |
| 53 | \( 1 + 0.267T + 53T^{2} \) |
| 59 | \( 1 - 11.4iT - 59T^{2} \) |
| 61 | \( 1 + 0.535T + 61T^{2} \) |
| 67 | \( 1 - 1.46iT - 67T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 + 9.46iT - 83T^{2} \) |
| 89 | \( 1 + 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 8.39iT - 97T^{2} \) |
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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
−7.77139306143961629822732876557, −7.34830887500170139093248588371, −6.77210902701531250532892507770, −5.84941242754529589314535899196, −5.28072999593660479153679792147, −4.42372900458056467662082465513, −3.97506642576963185699042054390, −2.76049507478416843875073529866, −1.19991264600636679015586514802, −0.46698173531477592581671371476,
1.13222105329963699464190322757, 2.08950087139785942106762745508, 3.05661034651306125954233263108, 3.66361966625167060608170690836, 4.82625416314967211969721486630, 5.41706139362196146515190430013, 6.08214175140015938558116170966, 6.79352433837296244084382057658, 7.85267949804141619713824758617, 8.356034466746222802947854315302