| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s − 3·11-s − 4·13-s + 15-s − 17-s − 19-s + 3·21-s + 4·23-s + 25-s + 27-s − 3·29-s + 8·31-s − 3·33-s + 3·35-s + 3·37-s − 4·39-s + 5·41-s + 8·43-s + 45-s + 9·47-s + 2·49-s − 51-s − 7·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 1.43·31-s − 0.522·33-s + 0.507·35-s + 0.493·37-s − 0.640·39-s + 0.780·41-s + 1.21·43-s + 0.149·45-s + 1.31·47-s + 2/7·49-s − 0.140·51-s − 0.961·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.821329680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821329680\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.260382618805256542430323786308, −7.83053371110021056834400314726, −7.17357628311824753957788925827, −6.24898778293344730746320092815, −5.17145210780284634244296896801, −4.85692765750778214058282261269, −3.88935816953954937855088642106, −2.53416210932782266200703139162, −2.31191132427133956795448473973, −0.952314738010447951173545884501,
0.952314738010447951173545884501, 2.31191132427133956795448473973, 2.53416210932782266200703139162, 3.88935816953954937855088642106, 4.85692765750778214058282261269, 5.17145210780284634244296896801, 6.24898778293344730746320092815, 7.17357628311824753957788925827, 7.83053371110021056834400314726, 8.260382618805256542430323786308
