| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 11-s + 13-s + 15-s − 2·17-s + 4·19-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s + 33-s + 4·35-s + 10·37-s − 39-s − 10·41-s + 4·43-s − 45-s + 9·49-s + 2·51-s + 10·53-s + 55-s − 4·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s + 0.676·35-s + 1.64·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s + 0.280·51-s + 1.37·53-s + 0.134·55-s − 0.529·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6747888862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6747888862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.99897628785129, −15.46823596350373, −15.02225232371058, −14.02377407158622, −13.59891100894075, −13.07187181820385, −12.46174357998330, −12.06121831475583, −11.45233318362287, −10.87134763337271, −10.10362647295153, −9.861078375148711, −9.171546990115352, −8.437975370013624, −7.824682579514006, −6.988558621651079, −6.700842257414838, −5.897502855074067, −5.528333676553616, −4.462701157342939, −4.044510787285396, −3.103597667149584, −2.685942815458907, −1.376090895638587, −0.3768695254336265,
0.3768695254336265, 1.376090895638587, 2.685942815458907, 3.103597667149584, 4.044510787285396, 4.462701157342939, 5.528333676553616, 5.897502855074067, 6.700842257414838, 6.988558621651079, 7.824682579514006, 8.437975370013624, 9.171546990115352, 9.861078375148711, 10.10362647295153, 10.87134763337271, 11.45233318362287, 12.06121831475583, 12.46174357998330, 13.07187181820385, 13.59891100894075, 14.02377407158622, 15.02225232371058, 15.46823596350373, 15.99897628785129
