| L(s) = 1 | + (0.617 + 1.60i)2-s + (0.747 − 1.56i)3-s + (−0.721 + 0.649i)4-s + (−1.46 − 1.69i)5-s + (2.97 + 0.238i)6-s + (−0.648 + 0.998i)7-s + (1.57 + 0.805i)8-s + (−1.88 − 2.33i)9-s + (1.82 − 3.39i)10-s + (2.93 − 1.54i)11-s + (0.475 + 1.61i)12-s + (2.43 − 3.00i)13-s + (−2.00 − 0.426i)14-s + (−3.73 + 1.01i)15-s + (−0.522 + 4.97i)16-s + (−0.736 − 4.64i)17-s + ⋯ |
| L(s) = 1 | + (0.436 + 1.13i)2-s + (0.431 − 0.901i)3-s + (−0.360 + 0.324i)4-s + (−0.653 − 0.757i)5-s + (1.21 + 0.0973i)6-s + (−0.245 + 0.377i)7-s + (0.558 + 0.284i)8-s + (−0.627 − 0.778i)9-s + (0.576 − 1.07i)10-s + (0.884 − 0.466i)11-s + (0.137 + 0.465i)12-s + (0.675 − 0.834i)13-s + (−0.536 − 0.114i)14-s + (−0.964 + 0.262i)15-s + (−0.130 + 1.24i)16-s + (−0.178 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96641 - 0.200716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96641 - 0.200716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.747 + 1.56i)T \) |
| 5 | \( 1 + (1.46 + 1.69i)T \) |
| 11 | \( 1 + (-2.93 + 1.54i)T \) |
| good | 2 | \( 1 + (-0.617 - 1.60i)T + (-1.48 + 1.33i)T^{2} \) |
| 7 | \( 1 + (0.648 - 0.998i)T + (-2.84 - 6.39i)T^{2} \) |
| 13 | \( 1 + (-2.43 + 3.00i)T + (-2.70 - 12.7i)T^{2} \) |
| 17 | \( 1 + (0.736 + 4.64i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.640 + 0.207i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0360 - 0.134i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.49 + 0.954i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (0.590 + 5.62i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-4.29 - 8.42i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (0.596 - 2.80i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.08 - 4.04i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (10.2 - 0.538i)T + (46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (-1.58 + 10.0i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-0.913 - 1.01i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.965 - 9.18i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-3.17 - 11.8i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.10 - 1.52i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (13.2 - 6.74i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (1.80 - 4.06i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.46 + 1.80i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 - 0.170T + 89T^{2} \) |
| 97 | \( 1 + (-7.06 + 2.71i)T + (72.0 - 64.9i)T^{2} \) |
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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
−11.41367572119592862373083488331, −9.671465396534539467934140144731, −8.577507982625151212685098164092, −8.161813690712323557175459959542, −7.20722306262252760239993753398, −6.34926615173627523478439965786, −5.57098869543266767301273397560, −4.36836125387055393735404884277, −3.03781414890868754864113660760, −1.12329311857801291759836320020,
1.92080891370658047089204958321, 3.31281957989003811237762113756, 3.87357375930620265867237490941, 4.59300191399736979089335790693, 6.37969697043572305095972626157, 7.34573346892774692097895994624, 8.522496676233627662766214404912, 9.503643512214422992840264741219, 10.46980356720211243321373737272, 10.85438018139725083892129420587
