L(s) = 1 | + (0.167 − 0.167i)2-s + (−0.707 + 0.707i)3-s + 1.94i·4-s + 0.236i·6-s + (2.64 − 0.0627i)7-s + (0.658 + 0.658i)8-s − 1.00i·9-s + 3.98·11-s + (−1.37 − 1.37i)12-s + (−0.500 + 0.500i)13-s + (0.431 − 0.452i)14-s − 3.66·16-s + (1.67 + 1.67i)17-s + (−0.167 − 0.167i)18-s − 7.21·19-s + ⋯ |
L(s) = 1 | + (0.118 − 0.118i)2-s + (−0.408 + 0.408i)3-s + 0.972i·4-s + 0.0964i·6-s + (0.999 − 0.0237i)7-s + (0.232 + 0.232i)8-s − 0.333i·9-s + 1.20·11-s + (−0.396 − 0.396i)12-s + (−0.138 + 0.138i)13-s + (0.115 − 0.120i)14-s − 0.917·16-s + (0.407 + 0.407i)17-s + (−0.0393 − 0.0393i)18-s − 1.65·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15619 + 0.937532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15619 + 0.937532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.64 + 0.0627i)T \) |
good | 2 | \( 1 + (-0.167 + 0.167i)T - 2iT^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + (0.500 - 0.500i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.67 - 1.67i)T + 17iT^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + (-5.16 - 5.16i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 - 4.93iT - 31T^{2} \) |
| 37 | \( 1 + (0.292 - 0.292i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (3.65 + 3.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.305 - 0.305i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.39 + 5.39i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 7.11iT - 61T^{2} \) |
| 67 | \( 1 + (0.944 - 0.944i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + (1.38 - 1.38i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.64iT - 79T^{2} \) |
| 83 | \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + (7.43 + 7.43i)T + 97iT^{2} \) |
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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
−11.16759761775196453324635994748, −10.42166381740154282152972994389, −9.012882911889267374225751107820, −8.588698286928173394952055909407, −7.41151790124950242472893664951, −6.57961457658482298180001689590, −5.22021545799015036017413613523, −4.29752432917559528673179465751, −3.46056031560316024364367422331, −1.76244600401334390775556571350,
0.990411593222280616200215389380, 2.20487769300152039450162614917, 4.29224259474778683965178709482, 5.01135343645831204109165775044, 6.17811949663577628544107085074, 6.72693644705053578474769519989, 7.935769884750667501159680983758, 8.904115519266648139667858070478, 9.861788725625197625946233750195, 10.90171704266034654258527465163