| L(s) = 1 | − 3-s + 3.08·5-s − 0.648·7-s + 9-s − 11-s − 13-s − 3.08·15-s + 1.73·17-s − 3.35·19-s + 0.648·21-s + 7.69·23-s + 4.52·25-s − 27-s + 9.14·29-s − 5.08·31-s + 33-s − 2·35-s − 1.29·37-s + 39-s − 5.69·41-s − 2.43·43-s + 3.08·45-s + 4.17·47-s − 6.58·49-s − 1.73·51-s + 5.46·53-s − 3.08·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.38·5-s − 0.244·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 0.796·15-s + 0.420·17-s − 0.768·19-s + 0.141·21-s + 1.60·23-s + 0.904·25-s − 0.192·27-s + 1.69·29-s − 0.913·31-s + 0.174·33-s − 0.338·35-s − 0.213·37-s + 0.160·39-s − 0.889·41-s − 0.371·43-s + 0.460·45-s + 0.608·47-s − 0.940·49-s − 0.242·51-s + 0.751·53-s − 0.416·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.074125567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074125567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 + 0.648T + 7T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 - 9.14T + 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 + 5.69T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 - 4.17T + 47T^{2} \) |
| 53 | \( 1 - 5.46T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 67 | \( 1 - 8.55T + 67T^{2} \) |
| 71 | \( 1 - 7.64T + 71T^{2} \) |
| 73 | \( 1 + 4.11T + 73T^{2} \) |
| 79 | \( 1 + 3.20T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−7.968651302721272843612883659189, −6.89379363077403344466526250559, −6.63933277786649454514259840440, −5.80239089512397967942352969981, −5.20642520461131002308077009481, −4.69119083206035792436040575864, −3.48886553691874850564137664180, −2.60954117344390265158089905053, −1.80433623712464973560688569973, −0.76868555394425876919103587583,
0.76868555394425876919103587583, 1.80433623712464973560688569973, 2.60954117344390265158089905053, 3.48886553691874850564137664180, 4.69119083206035792436040575864, 5.20642520461131002308077009481, 5.80239089512397967942352969981, 6.63933277786649454514259840440, 6.89379363077403344466526250559, 7.968651302721272843612883659189
