L(s) = 1 | + 3-s + 7-s + 9-s + 3·11-s + 13-s + 5·17-s + 2·19-s + 21-s − 3·23-s + 27-s + 4·31-s + 3·33-s − 37-s + 39-s + 9·41-s + 2·43-s + 8·47-s − 6·49-s + 5·51-s + 53-s + 2·57-s − 4·59-s − 3·61-s + 63-s − 16·67-s − 3·69-s + 15·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 1.21·17-s + 0.458·19-s + 0.218·21-s − 0.625·23-s + 0.192·27-s + 0.718·31-s + 0.522·33-s − 0.164·37-s + 0.160·39-s + 1.40·41-s + 0.304·43-s + 1.16·47-s − 6/7·49-s + 0.700·51-s + 0.137·53-s + 0.264·57-s − 0.520·59-s − 0.384·61-s + 0.125·63-s − 1.95·67-s − 0.361·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.108112413\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.108112413\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−15.07003362063532, −14.39889288442260, −13.99152520384711, −13.87186574995310, −12.93034889397283, −12.40832803972086, −11.96089140818085, −11.44823854459163, −10.74822809045591, −10.25763074262898, −9.485996261190734, −9.297320628692162, −8.553663063451395, −7.870964362377594, −7.676542566683048, −6.850591455966797, −6.214267495868331, −5.666035266985298, −4.915513275297473, −4.203339152606012, −3.684210459801626, −3.032544723789980, −2.253222542018293, −1.420601337224999, −0.8200453084962188,
0.8200453084962188, 1.420601337224999, 2.253222542018293, 3.032544723789980, 3.684210459801626, 4.203339152606012, 4.915513275297473, 5.666035266985298, 6.214267495868331, 6.850591455966797, 7.676542566683048, 7.870964362377594, 8.553663063451395, 9.297320628692162, 9.485996261190734, 10.25763074262898, 10.74822809045591, 11.44823854459163, 11.96089140818085, 12.40832803972086, 12.93034889397283, 13.87186574995310, 13.99152520384711, 14.39889288442260, 15.07003362063532