| L(s) = 1 | + 2-s + 4-s + (0.311 − 2.21i)5-s − 1.52·7-s + 8-s + (0.311 − 2.21i)10-s − 2.42i·11-s + (−3.59 − 0.311i)13-s − 1.52·14-s + 16-s − 0.903i·17-s − 4.90i·19-s + (0.311 − 2.21i)20-s − 2.42i·22-s − 2.90i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.139 − 0.990i)5-s − 0.576·7-s + 0.353·8-s + (0.0983 − 0.700i)10-s − 0.732i·11-s + (−0.996 − 0.0862i)13-s − 0.407·14-s + 0.250·16-s − 0.219i·17-s − 1.12i·19-s + (0.0695 − 0.495i)20-s − 0.517i·22-s − 0.605i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.983378230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.983378230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.311 + 2.21i)T \) |
| 13 | \( 1 + (3.59 + 0.311i)T \) |
| good | 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 2.42iT - 11T^{2} \) |
| 17 | \( 1 + 0.903iT - 17T^{2} \) |
| 19 | \( 1 + 4.90iT - 19T^{2} \) |
| 23 | \( 1 + 2.90iT - 23T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 - 0.755iT - 31T^{2} \) |
| 37 | \( 1 + 0.428T + 37T^{2} \) |
| 41 | \( 1 + 7.80iT - 41T^{2} \) |
| 43 | \( 1 - 7.18iT - 43T^{2} \) |
| 47 | \( 1 - 0.949T + 47T^{2} \) |
| 53 | \( 1 + 2.56iT - 53T^{2} \) |
| 59 | \( 1 - 2.13iT - 59T^{2} \) |
| 61 | \( 1 + 5.05T + 61T^{2} \) |
| 67 | \( 1 - 5.18T + 67T^{2} \) |
| 71 | \( 1 + 5.37iT - 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 8.62iT - 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−9.508774273723145777427132866212, −8.781494137384970629127274820823, −7.88700229814142218211075090528, −6.86530343077850915722746461924, −6.08838445216748348530692428072, −5.08853797807319630901498144274, −4.55346814028551947524791249994, −3.31260847983837240431545011079, −2.32890666671790196169399406661, −0.64536370915583424903280212727,
1.93447826655791077710393049118, 2.91875176971290947014847975816, 3.79834477444743545711282782808, 4.84825165428661630815397316422, 5.88065767700914399844201575707, 6.63565590545006351158984026935, 7.30595936947220909720754461370, 8.112390536408129724377364166782, 9.591567682778972033268499601605, 10.00373404627119349202933220124
