L(s) = 1 | + (−1.54 − 1.54i)2-s + (1.73 + 0.00622i)3-s + 2.76i·4-s + (−2.66 − 2.68i)6-s + (−0.707 + 0.707i)7-s + (1.18 − 1.18i)8-s + (2.99 + 0.0215i)9-s − 3.38i·11-s + (−0.0172 + 4.79i)12-s + (0.206 + 0.206i)13-s + 2.18·14-s + 1.87·16-s + (−0.167 − 0.167i)17-s + (−4.59 − 4.66i)18-s − 5.31i·19-s + ⋯ |
L(s) = 1 | + (−1.09 − 1.09i)2-s + (0.999 + 0.00359i)3-s + 1.38i·4-s + (−1.08 − 1.09i)6-s + (−0.267 + 0.267i)7-s + (0.419 − 0.419i)8-s + (0.999 + 0.00718i)9-s − 1.02i·11-s + (−0.00497 + 1.38i)12-s + (0.0573 + 0.0573i)13-s + 0.583·14-s + 0.467·16-s + (−0.0406 − 0.0406i)17-s + (−1.08 − 1.09i)18-s − 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.691513 - 0.876996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.691513 - 0.876996i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.73 - 0.00622i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (1.54 + 1.54i)T + 2iT^{2} \) |
| 11 | \( 1 + 3.38iT - 11T^{2} \) |
| 13 | \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.167 + 0.167i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.31iT - 19T^{2} \) |
| 23 | \( 1 + (-5.07 + 5.07i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.84T + 29T^{2} \) |
| 31 | \( 1 - 9.11T + 31T^{2} \) |
| 37 | \( 1 + (-5.27 + 5.27i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.0314iT - 41T^{2} \) |
| 43 | \( 1 + (-3.76 - 3.76i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.56 - 3.56i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.55 - 3.55i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 + (6.34 - 6.34i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.95iT - 71T^{2} \) |
| 73 | \( 1 + (8.61 + 8.61i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 + (3.88 - 3.88i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.00T + 89T^{2} \) |
| 97 | \( 1 + (2.26 - 2.26i)T - 97iT^{2} \) |
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
−10.61569540360616486610838619407, −9.495774205794162945945465240037, −9.026570599561806058617475252906, −8.369837938946322236458560581653, −7.44531799936556615579034432840, −6.18319199208820336606329158996, −4.48787574779911140062019519737, −3.07623269960313593546898293099, −2.56611613960241575088882195394, −0.952697459981378890041056595491,
1.48357862972854387416017842213, 3.19447363552114762954626976877, 4.51133382122944267063599731889, 5.96756443332669802374390727947, 7.01240577053917730529835145228, 7.59658755235925573026231355817, 8.321611517754157193287656584895, 9.273786221423852841086624588150, 9.812304981368578216160759582968, 10.50678705497768124850711216382