| L(s) = 1 | + 0.183i·3-s + i·5-s + 1.46·7-s + 2.96·9-s + 5.80i·11-s + 2.09i·13-s − 0.183·15-s − 17-s + 2.08i·19-s + 0.268i·21-s − 6.21·23-s − 25-s + 1.09i·27-s + 1.31i·29-s + 1.86·31-s + ⋯ |
| L(s) = 1 | + 0.105i·3-s + 0.447i·5-s + 0.553·7-s + 0.988·9-s + 1.74i·11-s + 0.579i·13-s − 0.0472·15-s − 0.242·17-s + 0.479i·19-s + 0.0585i·21-s − 1.29·23-s − 0.200·25-s + 0.210i·27-s + 0.243i·29-s + 0.334·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724004304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724004304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.183iT - 3T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 5.80iT - 11T^{2} \) |
| 13 | \( 1 - 2.09iT - 13T^{2} \) |
| 19 | \( 1 - 2.08iT - 19T^{2} \) |
| 23 | \( 1 + 6.21T + 23T^{2} \) |
| 29 | \( 1 - 1.31iT - 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + 3.34iT - 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 0.551iT - 53T^{2} \) |
| 59 | \( 1 + 8.97iT - 59T^{2} \) |
| 61 | \( 1 + 15.0iT - 61T^{2} \) |
| 67 | \( 1 + 6.68iT - 67T^{2} \) |
| 71 | \( 1 + 9.24T + 71T^{2} \) |
| 73 | \( 1 - 4.25T + 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 - 6.49iT - 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.416987492462300833814450214587, −7.975328116694046472662463133552, −7.70536146107665040564088729015, −6.76112043874866183717290023869, −6.23850831279345347093325215034, −4.83774774387046025576016975214, −4.50399673700498495457305186163, −3.59282088618058987005119930694, −2.16154787590797829950450264130, −1.61684689093083680476079235745,
0.55726575759015238514125766763, 1.59703445422101084889941179742, 2.82267624700321040904312602211, 3.87693709790311986107266661229, 4.60151580082503874209583125983, 5.57727063853851276211583353916, 6.15395723607314192767319330373, 7.16330500116271199942788272342, 7.994003998274448466616901413152, 8.466232232243305640064599675641
