| L(s) = 1 | − 3-s + 1.69i·5-s + (−2.56 − 0.662i)7-s + 9-s + 3.02i·11-s − 6.04i·13-s − 1.69i·15-s − 4.34i·17-s − 1.12·19-s + (2.56 + 0.662i)21-s + 3.02i·23-s + 2.12·25-s − 27-s + 2·29-s − 3.02i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.758i·5-s + (−0.968 − 0.250i)7-s + 0.333·9-s + 0.910i·11-s − 1.67i·13-s − 0.437i·15-s − 1.05i·17-s − 0.257·19-s + (0.558 + 0.144i)21-s + 0.629i·23-s + 0.424·25-s − 0.192·27-s + 0.371·29-s − 0.525i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.088027833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.088027833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (2.56 + 0.662i)T \) |
| good | 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 + 6.04iT - 13T^{2} \) |
| 17 | \( 1 + 4.34iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 3.02iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 7.73iT - 41T^{2} \) |
| 43 | \( 1 + 8.10iT - 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 9.43iT - 61T^{2} \) |
| 67 | \( 1 - 2.06iT - 67T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 + 3.39iT - 73T^{2} \) |
| 79 | \( 1 + 4.71iT - 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 7.73iT - 89T^{2} \) |
| 97 | \( 1 + 8.68iT - 97T^{2} \) |
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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
−9.917250561829860272752143353293, −8.915483577756359682593827923464, −7.56905701025221177273248268107, −7.20401459031477964055404828505, −6.28896000860352226025167950057, −5.52811665873643998212994601566, −4.51634717324006356244897783507, −3.33036838841329216239080138925, −2.56375626902082812126707286007, −0.67057495725587579823207856294,
0.900896108658626997398673603435, 2.34995531400563220483803751186, 3.77773525367131207825232594979, 4.48995842424915327374043780830, 5.61389357822136724136234429348, 6.31855275639410702315190507325, 6.91011986528721586399771082853, 8.238565608268941415679308380796, 8.917329640694148037234412806674, 9.505987378297126516277229721157
