L(s) = 1 | + (1.30 − 1.14i)3-s + (−0.164 − 0.284i)5-s + (−0.5 + 0.866i)7-s + (0.400 − 2.97i)9-s + (−0.664 + 1.15i)11-s + (−1.53 − 2.66i)13-s + (−0.539 − 0.183i)15-s + 7.35·17-s + 2.93·19-s + (0.335 + 1.69i)21-s + (−3.34 − 5.79i)23-s + (2.44 − 4.23i)25-s + (−2.86 − 4.33i)27-s + (3.88 − 6.72i)29-s + (−1.63 − 2.83i)31-s + ⋯ |
L(s) = 1 | + (0.752 − 0.658i)3-s + (−0.0735 − 0.127i)5-s + (−0.188 + 0.327i)7-s + (0.133 − 0.991i)9-s + (−0.200 + 0.347i)11-s + (−0.426 − 0.739i)13-s + (−0.139 − 0.0474i)15-s + 1.78·17-s + 0.673·19-s + (0.0732 + 0.370i)21-s + (−0.697 − 1.20i)23-s + (0.489 − 0.847i)25-s + (−0.552 − 0.833i)27-s + (0.720 − 1.24i)29-s + (−0.293 − 0.508i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.948538291\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.948538291\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 + 1.14i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.164 + 0.284i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.664 - 1.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.53 + 2.66i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7.35T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + (3.34 + 5.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.88 + 6.72i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.329T + 37T^{2} \) |
| 41 | \( 1 + (0.135 + 0.234i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.48 - 9.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.571 + 0.989i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 + (-0.372 - 0.644i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.42 - 7.65i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.28 - 7.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.60T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + (0.628 - 1.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.0316 - 0.0548i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-5.51 + 9.55i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.986183769993446895467171635388, −8.799457302156853025994683663389, −7.995165182638250427554117296378, −7.55695302995748684866261387600, −6.42608376166459075922207433109, −5.61263629115377492106811649710, −4.39560705230468956801808294129, −3.16200537274495010437109373348, −2.40984997072479603826990605551, −0.873282736670917357803788238461,
1.59768821872599677791134235527, 3.16723053330200251047595758924, 3.59688393453420917764806683674, 4.91301900421821281093291319273, 5.61334109586555264927880977765, 7.08795708857482909548177646409, 7.61946230131509330652176286098, 8.578379941677927568701620964899, 9.408911307024673306823225355852, 10.03451416320941426275088265751