L(s) = 1 | + 2.40·2-s + (−1.54 − 0.777i)3-s + 3.79·4-s + (−3.72 − 1.87i)6-s + (2.44 + i)7-s + 4.31·8-s + (1.79 + 2.40i)9-s − 4.31i·11-s + (−5.86 − 2.94i)12-s + 2.44·13-s + (5.89 + 2.40i)14-s + 2.79·16-s + 5.89i·17-s + (4.31 + 5.79i)18-s − 6.83i·19-s + ⋯ |
L(s) = 1 | + 1.70·2-s + (−0.893 − 0.448i)3-s + 1.89·4-s + (−1.52 − 0.763i)6-s + (0.925 + 0.377i)7-s + 1.52·8-s + (0.597 + 0.802i)9-s − 1.29i·11-s + (−1.69 − 0.850i)12-s + 0.679·13-s + (1.57 + 0.643i)14-s + 0.697·16-s + 1.42i·17-s + (1.01 + 1.36i)18-s − 1.56i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.04848 - 0.601170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.04848 - 0.601170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.54 + 0.777i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 11 | \( 1 + 4.31iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 5.89iT - 17T^{2} \) |
| 19 | \( 1 + 6.83iT - 19T^{2} \) |
| 23 | \( 1 + 0.502T + 23T^{2} \) |
| 29 | \( 1 - 0.502iT - 29T^{2} \) |
| 31 | \( 1 - 4.38iT - 31T^{2} \) |
| 37 | \( 1 - 0.582iT - 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 - 2.58iT - 43T^{2} \) |
| 47 | \( 1 - 5.89iT - 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.38iT - 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.31iT - 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 + 7.12iT - 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 0.511T + 97T^{2} \) |
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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
−11.04062793752575610481618287482, −10.78074906822906748721367209126, −8.800959400687237841241782575664, −7.84461732843784647386058911276, −6.56642636090376033403316221122, −6.01931463924588228080062830659, −5.18276511541666833203945997723, −4.37864517785703175601360960383, −3.07983745546653294547553423368, −1.59223454252716779021796001491,
1.80635054252704377120056768335, 3.56080262282435048760216865208, 4.46740661566677051155084280660, 5.04432419833091670097090958967, 5.95391106355519052356240461191, 6.91029335456469106493503287249, 7.77845709138040800277017547243, 9.469607367299762030124791161008, 10.44011571871915881264715211241, 11.23175953732212124800993744466