| L(s) = 1 | − 5-s + 4·11-s + 13-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 2·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s − 8·47-s − 7·49-s − 6·53-s − 4·55-s − 12·59-s − 2·61-s − 65-s + 4·67-s − 6·73-s − 16·79-s − 4·83-s + 2·85-s − 10·89-s − 4·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 49-s − 0.824·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s − 0.124·65-s + 0.488·67-s − 0.702·73-s − 1.80·79-s − 0.439·83-s + 0.216·85-s − 1.05·89-s − 0.410·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.322592688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322592688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.70207641696339484920839241991, −6.96803123102703029709093160118, −6.43695550060445450001023656796, −5.76579822409874646510496136587, −4.62981604092414037986499809056, −4.44347772636458903616706324481, −3.30362953635638720467719829301, −2.87000851906639316362986982057, −1.52071592036882198484496349519, −0.798842361418802905667250309202,
0.798842361418802905667250309202, 1.52071592036882198484496349519, 2.87000851906639316362986982057, 3.30362953635638720467719829301, 4.44347772636458903616706324481, 4.62981604092414037986499809056, 5.76579822409874646510496136587, 6.43695550060445450001023656796, 6.96803123102703029709093160118, 7.70207641696339484920839241991
