| L(s) = 1 | + 2.03i·5-s + 3.00i·7-s + 4.39·11-s + 4.79·13-s + 2.70i·17-s + i·19-s + 1.06·23-s + 0.866·25-s + 2.56i·29-s − 2.28i·31-s − 6.11·35-s + 4.52·37-s − 2.51i·41-s − 3.98i·43-s − 1.12·47-s + ⋯ |
| L(s) = 1 | + 0.909i·5-s + 1.13i·7-s + 1.32·11-s + 1.32·13-s + 0.656i·17-s + 0.229i·19-s + 0.222·23-s + 0.173·25-s + 0.476i·29-s − 0.410i·31-s − 1.03·35-s + 0.744·37-s − 0.392i·41-s − 0.608i·43-s − 0.163·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172851640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172851640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.03iT - 5T^{2} \) |
| 7 | \( 1 - 3.00iT - 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 - 2.70iT - 17T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 - 2.56iT - 29T^{2} \) |
| 31 | \( 1 + 2.28iT - 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 + 2.51iT - 41T^{2} \) |
| 43 | \( 1 + 3.98iT - 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 + 6.72iT - 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 1.75iT - 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 + 0.322T + 73T^{2} \) |
| 79 | \( 1 + 2.71iT - 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 + 3.49iT - 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−8.830880151770820387005271294734, −8.514005674772115342671015334687, −7.41407309951205910424986714510, −6.45110398076899755998983530028, −6.20094841015205834291888797048, −5.28921177073594250610432007481, −4.00957089688832358320132390213, −3.41475640728455104623721715993, −2.38281094224563893871921677111, −1.33899613282228462375889892826,
0.853643664312708448303020201557, 1.40711414730190298780087516954, 3.06958272695252628061213764715, 4.09189960585342803782010635826, 4.44903005222828657870908856315, 5.55230699784000512683221145148, 6.45954364500622559423758971823, 7.04593086877102273643268696507, 7.985643480532402650141009941695, 8.716283205695277763035830206305
