| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 6·13-s − 15-s + 2·17-s + 4·19-s + 4·21-s − 8·23-s + 25-s − 27-s + 6·29-s − 4·35-s − 6·39-s + 10·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s − 2·51-s + 10·53-s − 4·57-s − 6·61-s − 4·63-s + 6·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.676·35-s − 0.960·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.37·53-s − 0.529·57-s − 0.768·61-s − 0.503·63-s + 0.744·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839070426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839070426\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.51790195971231, −12.26992875260667, −11.63301730032313, −11.38239994352730, −10.62180064329087, −10.30178783058577, −9.962158693865583, −9.465780194349225, −9.122371787142985, −8.447166690366979, −8.046777416576191, −7.467495454950514, −6.754799008441772, −6.537171459910775, −5.928185832275466, −5.820570749732002, −5.265911653061599, −4.411546008130932, −4.002823155807242, −3.422138493357171, −3.047944415914572, −2.395903326163034, −1.571339528555159, −1.053323859464012, −0.4185994402125238,
0.4185994402125238, 1.053323859464012, 1.571339528555159, 2.395903326163034, 3.047944415914572, 3.422138493357171, 4.002823155807242, 4.411546008130932, 5.265911653061599, 5.820570749732002, 5.928185832275466, 6.537171459910775, 6.754799008441772, 7.467495454950514, 8.046777416576191, 8.447166690366979, 9.122371787142985, 9.465780194349225, 9.962158693865583, 10.30178783058577, 10.62180064329087, 11.38239994352730, 11.63301730032313, 12.26992875260667, 12.51790195971231
