L(s) = 1 | − 2.61·2-s − 1.70i·3-s + 4.82·4-s + 0.846i·5-s + 4.45i·6-s + 3.51i·7-s − 7.37·8-s + 0.0885·9-s − 2.21i·10-s + i·11-s − 8.23i·12-s + 0.0408·13-s − 9.19i·14-s + 1.44·15-s + 9.62·16-s + (0.793 + 4.04i)17-s + ⋯ |
L(s) = 1 | − 1.84·2-s − 0.985i·3-s + 2.41·4-s + 0.378i·5-s + 1.81i·6-s + 1.33i·7-s − 2.60·8-s + 0.0295·9-s − 0.699i·10-s + 0.301i·11-s − 2.37i·12-s + 0.0113·13-s − 2.45i·14-s + 0.373·15-s + 2.40·16-s + (0.192 + 0.981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.542120 + 0.0526609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542120 + 0.0526609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - iT \) |
| 17 | \( 1 + (-0.793 - 4.04i)T \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 1.70iT - 3T^{2} \) |
| 5 | \( 1 - 0.846iT - 5T^{2} \) |
| 7 | \( 1 - 3.51iT - 7T^{2} \) |
| 13 | \( 1 - 0.0408T + 13T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 - 2.40iT - 23T^{2} \) |
| 29 | \( 1 + 9.14iT - 29T^{2} \) |
| 31 | \( 1 - 2.81iT - 31T^{2} \) |
| 37 | \( 1 - 1.51iT - 37T^{2} \) |
| 41 | \( 1 - 2.95iT - 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 0.595T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 - 6.01iT - 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 16.6iT - 71T^{2} \) |
| 73 | \( 1 + 6.52iT - 73T^{2} \) |
| 79 | \( 1 + 3.93iT - 79T^{2} \) |
| 83 | \( 1 - 5.74T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 11.6iT - 97T^{2} \) |
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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
−12.05782412748592271145204324962, −11.72454225668220305284025521143, −10.36650759241920529582103574025, −9.529616793941690253077738216215, −8.509321887738304556904482325097, −7.70414245571219081732079420225, −6.81070225215119146739166021156, −5.82982515586056981164073922792, −2.72109713567822012604295537971, −1.51722658987686601780797286219,
1.01278856522464693889090598455, 3.35137442880925681546772360411, 5.00832760364910493883709486905, 6.90354008049675493991116274904, 7.59038149596026162430269809865, 8.867599587680352827276206820914, 9.567768686564644814365480726249, 10.40696420231762810781777226860, 10.92482079344570353093270017548, 12.05596479938982172422251879305