L(s) = 1 | − 2.47i·2-s + i·3-s − 4.13·4-s + (−0.164 + 2.23i)5-s + 2.47·6-s + 3.36i·7-s + 5.28i·8-s − 9-s + (5.52 + 0.406i)10-s + 3.04·11-s − 4.13i·12-s + 3.21i·13-s + 8.33·14-s + (−2.23 − 0.164i)15-s + 4.81·16-s − 3.46i·17-s + ⋯ |
L(s) = 1 | − 1.75i·2-s + 0.577i·3-s − 2.06·4-s + (−0.0733 + 0.997i)5-s + 1.01·6-s + 1.27i·7-s + 1.86i·8-s − 0.333·9-s + (1.74 + 0.128i)10-s + 0.916·11-s − 1.19i·12-s + 0.893i·13-s + 2.22·14-s + (−0.575 − 0.0423i)15-s + 1.20·16-s − 0.840i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04904 - 0.0385368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04904 - 0.0385368i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.164 - 2.23i)T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 2.47iT - 2T^{2} \) |
| 7 | \( 1 - 3.36iT - 7T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 13 | \( 1 - 3.21iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 23 | \( 1 - 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 8.97T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 + 0.438iT - 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 + 7.50iT - 43T^{2} \) |
| 47 | \( 1 + 6.31iT - 47T^{2} \) |
| 53 | \( 1 - 2.14iT - 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 7.09T + 61T^{2} \) |
| 67 | \( 1 - 13.3iT - 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 1.62iT - 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 + 8.82iT - 97T^{2} \) |
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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
−11.81025855847775496630002702690, −11.07382036405964914706335336194, −10.02880329643698755015348928348, −9.346140880216993154721453825944, −8.638714455483450840334222032028, −6.82523152446976273567864333056, −5.41128362953880422525437956931, −4.08834861489746970175837330626, −3.10558076354431072619246257588, −2.05357209764584840513181524648,
0.854719026914251657156814621388, 4.01057215104286459503467276531, 4.90801704281784937551365263185, 6.19507937991315408243222866849, 6.84671525932490409621233701300, 8.033035334759471336712142072770, 8.369091217144708925585211241831, 9.542750859498350037492606883790, 10.74267023895193203370389843637, 12.34437273355768715481351902639