L(s) = 1 | + 2.64i·2-s + (0.917 − 0.917i)3-s − 5.02·4-s + (1.81 − 1.30i)5-s + (2.43 + 2.43i)6-s − 0.112·7-s − 8.00i·8-s + 1.31i·9-s + (3.45 + 4.81i)10-s + (1.31 − 1.31i)11-s + (−4.60 + 4.60i)12-s − 0.297i·14-s + (0.470 − 2.86i)15-s + 11.1·16-s + (1.93 − 1.93i)17-s − 3.49·18-s + ⋯ |
L(s) = 1 | + 1.87i·2-s + (0.529 − 0.529i)3-s − 2.51·4-s + (0.812 − 0.583i)5-s + (0.992 + 0.992i)6-s − 0.0424·7-s − 2.83i·8-s + 0.439i·9-s + (1.09 + 1.52i)10-s + (0.395 − 0.395i)11-s + (−1.32 + 1.32i)12-s − 0.0795i·14-s + (0.121 − 0.738i)15-s + 2.79·16-s + (0.468 − 0.468i)17-s − 0.823·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40767 + 1.27267i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40767 + 1.27267i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.81 + 1.30i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.64iT - 2T^{2} \) |
| 3 | \( 1 + (-0.917 + 0.917i)T - 3iT^{2} \) |
| 7 | \( 1 + 0.112T + 7T^{2} \) |
| 11 | \( 1 + (-1.31 + 1.31i)T - 11iT^{2} \) |
| 17 | \( 1 + (-1.93 + 1.93i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.92 + 4.92i)T - 19iT^{2} \) |
| 23 | \( 1 + (-2.27 - 2.27i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.65iT - 29T^{2} \) |
| 31 | \( 1 + (0.624 + 0.624i)T + 31iT^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + (-3.83 - 3.83i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.75 - 2.75i)T + 43iT^{2} \) |
| 47 | \( 1 + 0.345T + 47T^{2} \) |
| 53 | \( 1 + (3.59 - 3.59i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.908 - 0.908i)T + 59iT^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 + 0.144iT - 67T^{2} \) |
| 71 | \( 1 + (3.87 + 3.87i)T + 71iT^{2} \) |
| 73 | \( 1 + 9.06iT - 73T^{2} \) |
| 79 | \( 1 - 15.1iT - 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + (0.402 + 0.402i)T + 89iT^{2} \) |
| 97 | \( 1 + 14.9iT - 97T^{2} \) |
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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
−9.720772786800470265084013416749, −9.213972099436833036143886851231, −8.536396548774237283709304501856, −7.67710797271602579222238744632, −7.07407461431135785998431397190, −6.14466812683148506030205797911, −5.27809243068815337809343024658, −4.68370469536453810147310553907, −3.04834580092530890253456249238, −1.15111953299883969595790158844,
1.29587105592671260920213878213, 2.45055657953710593929773694665, 3.36237050987210521723279981454, 4.00837158733227568238689047160, 5.20819704107895535377270527129, 6.30398843321332726888437834164, 7.80518873600507897631304647259, 8.934130480330269149249678132956, 9.519915592723884840423396470805, 10.04209503178728010669298221914