| L(s) = 1 | + 2·5-s + 7-s − 3·9-s + 4·11-s − 6·17-s + 8·23-s − 25-s + 10·29-s + 8·31-s + 2·35-s + 6·37-s + 6·41-s − 4·43-s − 6·45-s + 8·47-s + 49-s − 6·53-s + 8·55-s + 8·59-s − 10·61-s − 3·63-s + 4·67-s + 8·71-s − 2·73-s + 4·77-s + 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s − 1.45·17-s + 1.66·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 1.04·59-s − 1.28·61-s − 0.377·63-s + 0.488·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.806503093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.806503093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.94888859473425, −13.81115661348837, −13.11852967518183, −12.62258926665572, −11.92966497913506, −11.55622412192522, −11.07035135505813, −10.67203759897921, −9.946354980544928, −9.427378700699557, −8.997128576470677, −8.558185657821462, −8.141371684870519, −7.257904419978971, −6.679100728237650, −6.245861717275949, −5.939290821415040, −5.033304178830037, −4.669966172709630, −4.099193926574315, −3.184148498505776, −2.605373213089626, −2.166426507808984, −1.214626009037834, −0.7061185081017239,
0.7061185081017239, 1.214626009037834, 2.166426507808984, 2.605373213089626, 3.184148498505776, 4.099193926574315, 4.669966172709630, 5.033304178830037, 5.939290821415040, 6.245861717275949, 6.679100728237650, 7.257904419978971, 8.141371684870519, 8.558185657821462, 8.997128576470677, 9.427378700699557, 9.946354980544928, 10.67203759897921, 11.07035135505813, 11.55622412192522, 11.92966497913506, 12.62258926665572, 13.11852967518183, 13.81115661348837, 13.94888859473425
