| L(s) = 1 | + 1.73i·5-s + 1.73i·11-s − 3.46i·17-s + 2·19-s + 2.00·25-s − 9·29-s − 5·31-s + 10·37-s + 10.3i·41-s − 3.46i·43-s + 12·47-s + 9·53-s − 2.99·55-s − 9·59-s + 13.8i·67-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s + 0.522i·11-s − 0.840i·17-s + 0.458·19-s + 0.400·25-s − 1.67·29-s − 0.898·31-s + 1.64·37-s + 1.62i·41-s − 0.528i·43-s + 1.75·47-s + 1.23·53-s − 0.404·55-s − 1.17·59-s + 1.69i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.599107167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.599107167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.922805769723184275094165454926, −7.30242079572461138145011395745, −6.97862422849910674120208893947, −5.99221646179381375155257645569, −5.42451029678438750515486412429, −4.51421604574644646684376970045, −3.77193084854915222473909803013, −2.87349734061416486565478200306, −2.25358042614799358280280331150, −1.03810558623116636556249377159,
0.43812721032310367002082372579, 1.45501337233404181680203322840, 2.38946038519245983373218456130, 3.50563295897533306604851463727, 4.07290778153711235634192678815, 4.96626039315885766655515788357, 5.68338728388227704195333280308, 6.13237782380758189583492995173, 7.22674530021183876229445174046, 7.72347155552803628293197550886
