L(s) = 1 | − 3-s + 5-s + 9-s + 2·13-s − 15-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 10·37-s − 2·39-s + 2·41-s − 4·43-s + 45-s + 8·47-s − 7·49-s − 6·51-s − 2·53-s + 4·57-s + 8·59-s − 2·61-s + 2·65-s − 12·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s − 0.274·53-s + 0.529·57-s + 1.04·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.341664838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341664838\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−10.97163175619009206137205774068, −10.19227653538760520507936434046, −9.328467546447739299100058924123, −8.304308599781223866728910598449, −7.22612323207528452617340244852, −6.22207440469416781281899599638, −5.45564722344211377728395826291, −4.33436868576345517333214359577, −2.93279961256840981884560530804, −1.21912253617593317626144578411,
1.21912253617593317626144578411, 2.93279961256840981884560530804, 4.33436868576345517333214359577, 5.45564722344211377728395826291, 6.22207440469416781281899599638, 7.22612323207528452617340244852, 8.304308599781223866728910598449, 9.328467546447739299100058924123, 10.19227653538760520507936434046, 10.97163175619009206137205774068