| L(s) = 1 | + 10.5·5-s + 55.2·11-s − 83.5·13-s + 95.2·17-s − 83.5·19-s + 165.·23-s − 12.9·25-s − 110.·29-s + 83.5·31-s + 78·37-s + 412.·41-s + 148·43-s − 465.·47-s − 110.·53-s + 584.·55-s + 550.·59-s + 584.·61-s − 883.·65-s − 260·67-s + 718.·71-s − 668.·73-s + 664·79-s + 126.·83-s + 1.00e3·85-s + 878.·89-s − 883.·95-s − 1.16e3·97-s + ⋯ |
| L(s) = 1 | + 0.946·5-s + 1.51·11-s − 1.78·13-s + 1.35·17-s − 1.00·19-s + 1.50·23-s − 0.103·25-s − 0.707·29-s + 0.483·31-s + 0.346·37-s + 1.57·41-s + 0.524·43-s − 1.44·47-s − 0.286·53-s + 1.43·55-s + 1.21·59-s + 1.22·61-s − 1.68·65-s − 0.474·67-s + 1.20·71-s − 1.07·73-s + 0.945·79-s + 0.167·83-s + 1.28·85-s + 1.04·89-s − 0.954·95-s − 1.22·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.896225723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.896225723\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 10.5T + 125T^{2} \) |
| 11 | \( 1 - 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 83.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78T + 5.06e4T^{2} \) |
| 41 | \( 1 - 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 148T + 7.95e4T^{2} \) |
| 47 | \( 1 + 465.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 110.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 584.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 260T + 3.00e5T^{2} \) |
| 71 | \( 1 - 718.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 668.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 664T + 4.93e5T^{2} \) |
| 83 | \( 1 - 126.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 878.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−9.331909889320573797526876926913, −8.146993828085371579969024128642, −7.23099391180372710510931422991, −6.56245256709276649577392007100, −5.71244660438377810869209544165, −4.92771736151253921436304491238, −3.96740563076351451566209528488, −2.80921202527105905131872543715, −1.88968294103218296450199345763, −0.826405437379676502600486786119,
0.826405437379676502600486786119, 1.88968294103218296450199345763, 2.80921202527105905131872543715, 3.96740563076351451566209528488, 4.92771736151253921436304491238, 5.71244660438377810869209544165, 6.56245256709276649577392007100, 7.23099391180372710510931422991, 8.146993828085371579969024128642, 9.331909889320573797526876926913
