L(s) = 1 | − 2.76i·2-s − 5.62·4-s − 1.86i·7-s + 10.0i·8-s − 11-s + 4.62i·13-s − 5.14·14-s + 16.4·16-s + 2.49i·17-s + 5.38·19-s + 2.76i·22-s − 7.14i·23-s + 12.7·26-s + 10.4i·28-s − 3.52·29-s + ⋯ |
L(s) = 1 | − 1.95i·2-s − 2.81·4-s − 0.704i·7-s + 3.54i·8-s − 0.301·11-s + 1.28i·13-s − 1.37·14-s + 4.10·16-s + 0.604i·17-s + 1.23·19-s + 0.588i·22-s − 1.49i·23-s + 2.50·26-s + 1.98i·28-s − 0.654·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.336580326\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.336580326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.76iT - 2T^{2} \) |
| 7 | \( 1 + 1.86iT - 7T^{2} \) |
| 13 | \( 1 - 4.62iT - 13T^{2} \) |
| 17 | \( 1 - 2.49iT - 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 7.14iT - 23T^{2} \) |
| 29 | \( 1 + 3.52T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 - 8.87iT - 37T^{2} \) |
| 41 | \( 1 - 0.761T + 41T^{2} \) |
| 43 | \( 1 + 7.40iT - 43T^{2} \) |
| 47 | \( 1 + 0.373iT - 47T^{2} \) |
| 53 | \( 1 - 5.45iT - 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.77iT - 73T^{2} \) |
| 79 | \( 1 - 6.01T + 79T^{2} \) |
| 83 | \( 1 + 14.5iT - 83T^{2} \) |
| 89 | \( 1 - 9.04T + 89T^{2} \) |
| 97 | \( 1 + 16.3iT - 97T^{2} \) |
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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
−8.848401824437264712022487223449, −8.255912817364448250404784452303, −7.24126010964467134052849166505, −6.09216054140344809363317340962, −4.89250653024869579753250941629, −4.37750956144432272704058928501, −3.57974394217545696398988694457, −2.67903730638321151792386517824, −1.72038707999507842265103927209, −0.70377265828950853634128280673,
0.818466543822623091284315871820, 2.92502133364387229471392122381, 3.88748490994798439627027479673, 5.11067138495709330918360779162, 5.43515068853693070366077472581, 6.05577297563978035319610800562, 7.08675836860646258605325479863, 7.69468948071611929163166483353, 8.151376952238876324373690723834, 9.085591297245701595363814208045