| L(s) = 1 | + 2·5-s − 7-s + 11-s − 6·13-s + 4·17-s + 8·19-s + 6·23-s − 25-s − 2·29-s + 4·31-s − 2·35-s − 37-s − 7·41-s − 2·43-s + 9·47-s − 6·49-s + 3·53-s + 2·55-s − 12·59-s + 4·61-s − 12·65-s + 7·71-s + 7·73-s − 77-s + 3·83-s + 8·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.301·11-s − 1.66·13-s + 0.970·17-s + 1.83·19-s + 1.25·23-s − 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.338·35-s − 0.164·37-s − 1.09·41-s − 0.304·43-s + 1.31·47-s − 6/7·49-s + 0.412·53-s + 0.269·55-s − 1.56·59-s + 0.512·61-s − 1.48·65-s + 0.830·71-s + 0.819·73-s − 0.113·77-s + 0.329·83-s + 0.867·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.292978806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.292978806\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.070458782324392630483813369831, −7.36158007054405283647547491277, −6.86956114686397304905880369887, −5.93411476803816462364067798961, −5.26249447883632910171765768247, −4.79513810980139595130903330602, −3.46408054935867786603187582223, −2.88308379745334425392558663656, −1.90541685045441435373462383488, −0.830738487980047601815971212772,
0.830738487980047601815971212772, 1.90541685045441435373462383488, 2.88308379745334425392558663656, 3.46408054935867786603187582223, 4.79513810980139595130903330602, 5.26249447883632910171765768247, 5.93411476803816462364067798961, 6.86956114686397304905880369887, 7.36158007054405283647547491277, 8.070458782324392630483813369831
