| L(s) = 1 | − i·2-s − 2.44i·3-s − 4-s − 2.44·6-s − 4.44i·7-s + i·8-s − 2.99·9-s + 2·11-s + 2.44i·12-s − 2.44i·13-s − 4.44·14-s + 16-s + 2i·17-s + 2.99i·18-s − 6.44·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.41i·3-s − 0.5·4-s − 0.999·6-s − 1.68i·7-s + 0.353i·8-s − 0.999·9-s + 0.603·11-s + 0.707i·12-s − 0.679i·13-s − 1.18·14-s + 0.250·16-s + 0.485i·17-s + 0.707i·18-s − 1.47·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223306133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223306133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 7 | \( 1 + 4.44iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 2.44iT - 23T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 3.44iT - 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8.89iT - 43T^{2} \) |
| 47 | \( 1 + 5.89iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 + 7.34iT - 97T^{2} \) |
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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
−8.830299081497925628418725679171, −8.107545705653241098111260373837, −7.34411906545359516540771750388, −6.71278389644999100581948818429, −5.87900272379343897289268249127, −4.41697287223957078050022296075, −3.77535928326288717282233463683, −2.45422432814042519338539062846, −1.38491252317642979251150679409, −0.51444890166230282369496532720,
2.16217307240048436926906082695, 3.41128864377746687933567847092, 4.39403923185426901881049264464, 5.01350637095519203066564162002, 5.93906653840011045964252745883, 6.51261255967793713650543328803, 7.83477388028832628104098747034, 8.746962839106522726473967911899, 9.341081100002142103787623386390, 9.527549575094134050406266131161
