L(s) = 1 | + (0.869 + 1.11i)2-s + 3-s + (−0.489 + 1.93i)4-s − 1.54·5-s + (0.869 + 1.11i)6-s − 2.92·7-s + (−2.58 + 1.13i)8-s + 9-s + (−1.34 − 1.72i)10-s + 4.46i·11-s + (−0.489 + 1.93i)12-s + 2.71i·13-s + (−2.54 − 3.26i)14-s − 1.54·15-s + (−3.52 − 1.89i)16-s + 0.363i·17-s + ⋯ |
L(s) = 1 | + (0.614 + 0.788i)2-s + 0.577·3-s + (−0.244 + 0.969i)4-s − 0.689·5-s + (0.354 + 0.455i)6-s − 1.10·7-s + (−0.915 + 0.402i)8-s + 0.333·9-s + (−0.423 − 0.544i)10-s + 1.34i·11-s + (−0.141 + 0.559i)12-s + 0.753i·13-s + (−0.679 − 0.871i)14-s − 0.398·15-s + (−0.880 − 0.474i)16-s + 0.0881i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.309206 + 1.46795i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.309206 + 1.46795i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.869 - 1.11i)T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + (-3.23 - 3.53i)T \) |
good | 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 - 4.46iT - 11T^{2} \) |
| 13 | \( 1 - 2.71iT - 13T^{2} \) |
| 17 | \( 1 - 0.363iT - 17T^{2} \) |
| 19 | \( 1 - 1.20iT - 19T^{2} \) |
| 29 | \( 1 + 3.26iT - 29T^{2} \) |
| 31 | \( 1 - 1.19iT - 31T^{2} \) |
| 37 | \( 1 - 9.28T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 8.22iT - 43T^{2} \) |
| 47 | \( 1 + 2.10iT - 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 5.63T + 61T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 6.80iT - 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 - 16.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.99iT - 89T^{2} \) |
| 97 | \( 1 + 13.6iT - 97T^{2} \) |
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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
−11.50423287658638463778240890761, −9.935377089526644742321100192075, −9.330921540157526878569055411850, −8.332688114460819635062019313447, −7.30777738626023567638284841824, −6.91751593897436579968137946579, −5.70583909379459207519022717671, −4.34224904295957616010575942031, −3.76089631391061881255344160732, −2.49318045053860651395400200142,
0.65500498495904058838841093339, 2.79132703643675834931187039166, 3.33603582703481833006500797140, 4.35457416920211208697367577378, 5.70432376557593519998679273063, 6.56051086302847461469125239591, 7.84688408578373380039657458779, 8.838291564586559229351036261308, 9.597595715763976298717487332190, 10.56612371020416174250438636805