| L(s) = 1 | − 1.73i·3-s − 2.99·9-s + 2·13-s − 3.46i·19-s + 5·25-s + 5.19i·27-s − 10.3i·31-s − 10·37-s − 3.46i·39-s − 10.3i·43-s − 5.99·57-s − 14·61-s + 3.46i·67-s − 10·73-s − 8.66i·75-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s − 0.999·9-s + 0.554·13-s − 0.794i·19-s + 25-s + 0.999i·27-s − 1.86i·31-s − 1.64·37-s − 0.554i·39-s − 1.58i·43-s − 0.794·57-s − 1.79·61-s + 0.423i·67-s − 1.17·73-s − 0.999i·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208081799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208081799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.763354738188187608292141215526, −7.76077986013522224737288385081, −7.16536976650056772683255048129, −6.40303506450071400686339200771, −5.69174789907543695266489746142, −4.77666763692699261946312198812, −3.60181559733772757475246217339, −2.64928102845260039095935617264, −1.65679254041983229888003939170, −0.41531380651512062279424756198,
1.43986159162800746822821084039, 2.93526184693426707344651788613, 3.56668456363303578500709839870, 4.56068834799076308527594398306, 5.23641759229527618688331756827, 6.09547455772947283310479794247, 6.89421756484277699151705267606, 7.992907980026055313790914412330, 8.685604204031375135554107417137, 9.216816332137165395494573315500
