| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 2·19-s − 3·21-s + 3·23-s + 25-s + 27-s − 2·29-s + 6·31-s + 33-s − 3·35-s + 11·37-s − 39-s − 5·41-s − 4·43-s + 45-s + 10·47-s + 2·49-s − 51-s + 11·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.654·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s + 0.174·33-s − 0.507·35-s + 1.80·37-s − 0.160·39-s − 0.780·41-s − 0.609·43-s + 0.149·45-s + 1.45·47-s + 2/7·49-s − 0.140·51-s + 1.51·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245193824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245193824\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.859742629128846146336520631559, −7.972551023048584427904151580263, −7.11315490900767649931627865439, −6.52098216309501079985355368795, −5.76684274070239466860561733520, −4.77834348392706889939521142886, −3.83066202020499030341074962519, −3.00135498973141972833429935360, −2.26942722047331000485028011206, −0.888327107629161525168347183801,
0.888327107629161525168347183801, 2.26942722047331000485028011206, 3.00135498973141972833429935360, 3.83066202020499030341074962519, 4.77834348392706889939521142886, 5.76684274070239466860561733520, 6.52098216309501079985355368795, 7.11315490900767649931627865439, 7.972551023048584427904151580263, 8.859742629128846146336520631559
