| L(s) = 1 | + 5-s − 7-s + 11-s + 2·13-s + 6·19-s − 23-s − 4·25-s − 7·31-s − 35-s + 5·37-s − 6·41-s + 2·43-s − 4·47-s + 49-s + 10·53-s + 55-s − 13·59-s − 4·61-s + 2·65-s + 9·67-s + 9·71-s − 14·73-s − 77-s + 6·79-s + 2·83-s + 89-s − 2·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.554·13-s + 1.37·19-s − 0.208·23-s − 4/5·25-s − 1.25·31-s − 0.169·35-s + 0.821·37-s − 0.937·41-s + 0.304·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.134·55-s − 1.69·59-s − 0.512·61-s + 0.248·65-s + 1.09·67-s + 1.06·71-s − 1.63·73-s − 0.113·77-s + 0.675·79-s + 0.219·83-s + 0.105·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.91311994308530, −14.33727600128050, −13.77088632138924, −13.48630294063245, −12.96112931660680, −12.29473206344387, −11.83605352511786, −11.31833915287632, −10.75900129343886, −10.15818362247688, −9.586443216490491, −9.282061506635190, −8.703081619008125, −7.896568705143786, −7.564424523879429, −6.768688936626435, −6.358785554080853, −5.574129193076256, −5.393600866266527, −4.438128838022570, −3.765714807681445, −3.287879468178840, −2.518462964248277, −1.714855521076413, −1.082368808332896, 0,
1.082368808332896, 1.714855521076413, 2.518462964248277, 3.287879468178840, 3.765714807681445, 4.438128838022570, 5.393600866266527, 5.574129193076256, 6.358785554080853, 6.768688936626435, 7.564424523879429, 7.896568705143786, 8.703081619008125, 9.282061506635190, 9.586443216490491, 10.15818362247688, 10.75900129343886, 11.31833915287632, 11.83605352511786, 12.29473206344387, 12.96112931660680, 13.48630294063245, 13.77088632138924, 14.33727600128050, 14.91311994308530
