| L(s) = 1 | + 5-s − 2·11-s + 2·13-s − 2·17-s + 8·19-s − 4·23-s + 25-s + 10·29-s + 2·31-s − 37-s + 8·41-s + 6·43-s + 6·47-s − 7·49-s − 2·55-s + 4·59-s + 4·61-s + 2·65-s + 16·67-s + 4·71-s + 6·73-s + 10·79-s − 16·83-s − 2·85-s + 2·89-s + 8·95-s − 8·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s + 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.359·31-s − 0.164·37-s + 1.24·41-s + 0.914·43-s + 0.875·47-s − 49-s − 0.269·55-s + 0.520·59-s + 0.512·61-s + 0.248·65-s + 1.95·67-s + 0.474·71-s + 0.702·73-s + 1.12·79-s − 1.75·83-s − 0.216·85-s + 0.211·89-s + 0.820·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.588313996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.588313996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.88226874406679, −13.14910159027147, −12.83986166441408, −12.15635527020311, −11.84534484272963, −11.14264963170395, −10.82462928181575, −10.20294710272378, −9.684027999462960, −9.461194572966239, −8.643908403059289, −8.252471155246410, −7.770214106351741, −7.137287568015271, −6.649234652302410, −6.012539909872611, −5.620227789445946, −5.017143148710492, −4.506487391642627, −3.807815411676311, −3.183017880356816, −2.566576890120958, −2.083213420932357, −1.083284871214660, −0.6873795017548569,
0.6873795017548569, 1.083284871214660, 2.083213420932357, 2.566576890120958, 3.183017880356816, 3.807815411676311, 4.506487391642627, 5.017143148710492, 5.620227789445946, 6.012539909872611, 6.649234652302410, 7.137287568015271, 7.770214106351741, 8.252471155246410, 8.643908403059289, 9.461194572966239, 9.684027999462960, 10.20294710272378, 10.82462928181575, 11.14264963170395, 11.84534484272963, 12.15635527020311, 12.83986166441408, 13.14910159027147, 13.88226874406679
