| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 5.65·11-s − 2·13-s + 15-s − 3.65·17-s + 5.65·19-s − 21-s − 5.65·23-s + 25-s − 27-s − 3.65·29-s + 4·31-s + 5.65·33-s − 35-s − 11.6·37-s + 2·39-s + 2·41-s − 1.65·43-s − 45-s − 2.34·47-s + 49-s + 3.65·51-s + 3.65·53-s + 5.65·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.70·11-s − 0.554·13-s + 0.258·15-s − 0.886·17-s + 1.29·19-s − 0.218·21-s − 1.17·23-s + 0.200·25-s − 0.192·27-s − 0.679·29-s + 0.718·31-s + 0.984·33-s − 0.169·35-s − 1.91·37-s + 0.320·39-s + 0.312·41-s − 0.252·43-s − 0.149·45-s − 0.341·47-s + 0.142·49-s + 0.512·51-s + 0.502·53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7320491065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7320491065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + 2.34T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.953056124101357801219717123241, −7.30503799426494971263094718696, −6.70987150077684420608156129207, −5.60159504054638159408297301681, −5.23370760531694699447169231722, −4.55284017186599483225658868334, −3.65360593517275508357171536654, −2.68257461223782511830123581662, −1.84998376772649747350168218787, −0.43415504156937784028334735364,
0.43415504156937784028334735364, 1.84998376772649747350168218787, 2.68257461223782511830123581662, 3.65360593517275508357171536654, 4.55284017186599483225658868334, 5.23370760531694699447169231722, 5.60159504054638159408297301681, 6.70987150077684420608156129207, 7.30503799426494971263094718696, 7.953056124101357801219717123241
