| L(s) = 1 | + 5i·7-s + 5i·13-s − 19-s + 7·31-s + 10i·37-s − 5i·43-s − 18·49-s − 13·61-s + 5i·67-s − 10i·73-s − 4·79-s − 25·91-s − 5i·97-s − 20i·103-s + 19·109-s + ⋯ |
| L(s) = 1 | + 1.88i·7-s + 1.38i·13-s − 0.229·19-s + 1.25·31-s + 1.64i·37-s − 0.762i·43-s − 2.57·49-s − 1.66·61-s + 0.610i·67-s − 1.17i·73-s − 0.450·79-s − 2.62·91-s − 0.507i·97-s − 1.97i·103-s + 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281663470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281663470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
| 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
−8.717333090204791611798453361056, −8.458977922491354144463185872546, −7.37925041713015986535745687257, −6.39302450206883075958968155699, −6.06265219306703355665865262601, −5.03745612959033359564318205548, −4.45670430796568630472806689582, −3.21389631703215382989902552813, −2.41260182677136107623503162583, −1.60010398080805915820508866624,
0.39229657431660952652605236117, 1.28177264963244280190096979497, 2.73224647575575663369456357834, 3.61871279566415536413523222071, 4.31029782917434790029497105093, 5.10729334517241851998332098234, 6.09750651855915398744012837930, 6.81531475745839131249806414152, 7.72155833311923291907485731016, 7.87112260524321714467039080242
