| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 2·13-s + 16-s − 2·17-s − 8·19-s − 2·22-s − 4·23-s + 2·26-s + 10·29-s + 2·31-s + 32-s − 2·34-s − 37-s − 8·38-s − 8·41-s + 6·43-s − 2·44-s − 4·46-s + 6·47-s − 7·49-s + 2·52-s + 10·58-s + 4·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 1.83·19-s − 0.426·22-s − 0.834·23-s + 0.392·26-s + 1.85·29-s + 0.359·31-s + 0.176·32-s − 0.342·34-s − 0.164·37-s − 1.29·38-s − 1.24·41-s + 0.914·43-s − 0.301·44-s − 0.589·46-s + 0.875·47-s − 49-s + 0.277·52-s + 1.31·58-s + 0.520·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.943638935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.943638935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 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
−15.74633064089666, −15.45664267469060, −14.76637904744817, −14.20859249720421, −13.66538417096152, −13.19187904915730, −12.62970149681363, −12.12621857365462, −11.55214425245331, −10.72406259652711, −10.54643072007397, −9.886278382392807, −8.957009541674282, −8.342693910359413, −8.011547881516757, −7.053704075681744, −6.390416345797452, −6.144643116159874, −5.198630612898709, −4.593459118782378, −4.054454974012280, −3.272198830508293, −2.428451196302414, −1.892565936643506, −0.6320821800644515,
0.6320821800644515, 1.892565936643506, 2.428451196302414, 3.272198830508293, 4.054454974012280, 4.593459118782378, 5.198630612898709, 6.144643116159874, 6.390416345797452, 7.053704075681744, 8.011547881516757, 8.342693910359413, 8.957009541674282, 9.886278382392807, 10.54643072007397, 10.72406259652711, 11.55214425245331, 12.12621857365462, 12.62970149681363, 13.19187904915730, 13.66538417096152, 14.20859249720421, 14.76637904744817, 15.45664267469060, 15.74633064089666
