L(s) = 1 | + (1.27 − 0.618i)2-s + (2.16 + 2.16i)3-s + (1.23 − 1.57i)4-s + (4.09 + 1.41i)6-s − 3.30·7-s + (0.594 − 2.76i)8-s + 6.40i·9-s + (2.01 + 2.01i)11-s + (6.08 − 0.738i)12-s + (0.794 + 0.794i)13-s + (−4.20 + 2.04i)14-s + (−0.955 − 3.88i)16-s − 4.61i·17-s + (3.96 + 8.14i)18-s + (3.48 − 3.48i)19-s + ⋯ |
L(s) = 1 | + (0.899 − 0.437i)2-s + (1.25 + 1.25i)3-s + (0.616 − 0.787i)4-s + (1.67 + 0.577i)6-s − 1.24·7-s + (0.210 − 0.977i)8-s + 2.13i·9-s + (0.606 + 0.606i)11-s + (1.75 − 0.213i)12-s + (0.220 + 0.220i)13-s + (−1.12 + 0.546i)14-s + (−0.238 − 0.971i)16-s − 1.11i·17-s + (0.934 + 1.91i)18-s + (0.800 − 0.800i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.96848 + 0.467192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.96848 + 0.467192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.27 + 0.618i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-2.16 - 2.16i)T + 3iT^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 + (-2.01 - 2.01i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.794 - 0.794i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.61iT - 17T^{2} \) |
| 19 | \( 1 + (-3.48 + 3.48i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.99T + 23T^{2} \) |
| 29 | \( 1 + (-1.95 + 1.95i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + (-0.448 + 0.448i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.02iT - 41T^{2} \) |
| 43 | \( 1 + (4.97 - 4.97i)T - 43iT^{2} \) |
| 47 | \( 1 - 5.49iT - 47T^{2} \) |
| 53 | \( 1 + (-3.35 + 3.35i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.07 + 2.07i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.557 - 0.557i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.636 - 0.636i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.85iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + (-9.48 - 9.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.62iT - 89T^{2} \) |
| 97 | \( 1 + 0.709iT - 97T^{2} \) |
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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
−11.31125148714538002366216561882, −10.13735334812526934661492275631, −9.639566840214335904993118344114, −9.095114781674463860716077750500, −7.54784266895609612606512296294, −6.44949536725779312571807039668, −5.08528902140475990726359051394, −4.08153751214928444580371512457, −3.34360283336008409519557243700, −2.36974160826004477895396514403,
1.85705023054726397453900987924, 3.29133121557239768290243004036, 3.71692419070275696612427063745, 5.94576000320103875121132741611, 6.41800843205425574034433286624, 7.43467275038315830741476668179, 8.218774655020555174473657848928, 9.009681366202350317243320076879, 10.24413321414476070483425837214, 11.84867044430255251644201472424