| L(s) = 1 | + 2i·7-s + 3·9-s − 4·11-s − 4i·13-s − 6i·17-s − 19-s − 2i·23-s + 6·29-s − 8·31-s − 4i·37-s + 6·41-s − 6i·43-s − 6i·47-s + 3·49-s + 8i·53-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s + 9-s − 1.20·11-s − 1.10i·13-s − 1.45i·17-s − 0.229·19-s − 0.417i·23-s + 1.11·29-s − 1.43·31-s − 0.657i·37-s + 0.937·41-s − 0.914i·43-s − 0.875i·47-s + 0.428·49-s + 1.09i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472989491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472989491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−9.077293093414878039845065090700, −8.274552119508155899957419576267, −7.47898526201407817200021756921, −6.88673264989188868325553981914, −5.58196388526139325212439436531, −5.23966529840681822352569847175, −4.17357629543407708518549710199, −2.93964007791977598814731013201, −2.23650415450181533851702708922, −0.57436921608937453122972389544,
1.26536922468714984063166449458, 2.31587808798972415861147701726, 3.72864693903098481959129775750, 4.31132258830550929197656182777, 5.22957222326122652244461581748, 6.32249610706455623820169931528, 7.02282302209655139132229049351, 7.75649600970488012689430024740, 8.454940702545327898411804806210, 9.484368424030761845389022164638
