| L(s) = 1 | + i·3-s + (1 + 2i)5-s − 4i·7-s − 9-s − 4·11-s + (−2 + i)15-s − 4i·17-s + 4·21-s + 4i·23-s + (−3 + 4i)25-s − i·27-s + 6·29-s + 4·31-s − 4i·33-s + (8 − 4i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.447 + 0.894i)5-s − 1.51i·7-s − 0.333·9-s − 1.20·11-s + (−0.516 + 0.258i)15-s − 0.970i·17-s + 0.872·21-s + 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.192i·27-s + 1.11·29-s + 0.718·31-s − 0.696i·33-s + (1.35 − 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.863408 + 0.203823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.863408 + 0.203823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−15.20632694967217143748283721071, −13.93215941391317470745755448236, −13.42499414764942110476843074071, −11.46631660381950947840341031494, −10.41140256932189427651111766940, −9.862031298307833742750515813661, −7.88773361047321821012866564938, −6.70923995105228769594070884168, −4.92017918035329438398648872326, −3.16612952657833741339887507090,
2.33534661767996088087914166220, 5.11991543003276996574657604229, 6.16189212507377429778945636624, 8.145420643870246352308910942716, 8.890680176421272534017816198129, 10.39355477557200835705975196955, 12.10184840874469074539954273416, 12.64154176447702888313696528389, 13.69483881127454116103742377418, 15.13529243314735464060640336789
