L(s) = 1 | + (0.193 − 0.334i)3-s + 3.17i·5-s + (−2.98 + 1.72i)7-s + (1.42 + 2.46i)9-s + (2.98 + 1.72i)11-s + (−0.362 − 3.58i)13-s + (1.06 + 0.613i)15-s + (−0.886 − 1.53i)17-s + (2.88 − 1.66i)19-s + 1.33i·21-s + (0.193 − 0.334i)23-s − 5.07·25-s + 2.26·27-s + (2.28 − 3.96i)29-s + 11.0i·31-s + ⋯ |
L(s) = 1 | + (0.111 − 0.193i)3-s + 1.41i·5-s + (−1.12 + 0.650i)7-s + (0.475 + 0.822i)9-s + (0.898 + 0.518i)11-s + (−0.100 − 0.994i)13-s + (0.274 + 0.158i)15-s + (−0.215 − 0.372i)17-s + (0.661 − 0.382i)19-s + 0.290i·21-s + (0.0402 − 0.0697i)23-s − 1.01·25-s + 0.435·27-s + (0.424 − 0.735i)29-s + 1.98i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.962600 + 0.631208i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.962600 + 0.631208i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (0.362 + 3.58i)T \) |
good | 3 | \( 1 + (-0.193 + 0.334i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 3.17iT - 5T^{2} \) |
| 7 | \( 1 + (2.98 - 1.72i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.98 - 1.72i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.886 + 1.53i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.88 + 1.66i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.193 + 0.334i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.28 + 3.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 11.0iT - 31T^{2} \) |
| 37 | \( 1 + (1.40 + 0.810i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.96 + 1.13i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.36 + 9.29i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 8.11iT - 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + (-5.47 + 3.16i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.21 + 2.09i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.91 - 4.57i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.88 + 5.12i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 8.40iT - 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 - 1.11iT - 83T^{2} \) |
| 89 | \( 1 + (15.4 + 8.91i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.88 + 2.24i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.55479456768813631251895898461, −11.64709163484843525886256330883, −10.40809460914211047462076511367, −9.913236358691717243486011498882, −8.609835040578601814292960492275, −7.11629853739326235304258856928, −6.75427000183872454866379814863, −5.33106156504772528924087079571, −3.48177403081796710384908377767, −2.45405001847680790829964400016,
1.09152126373637792741777817710, 3.63697916152885676023342731551, 4.43548036402817540838517116076, 6.04733843940162153577053456352, 6.97177017377193432678477924863, 8.477569518940819106645149523066, 9.413059907123763526516593409649, 9.806634825764168657283794710602, 11.44749990227751836661872392890, 12.32765414326599675911928031798