| L(s) = 1 | + 3·3-s + 13·7-s + 9·9-s + 6·11-s − 5·13-s + 78·17-s + 65·19-s + 39·21-s − 138·23-s + 27·27-s + 66·29-s + 299·31-s + 18·33-s + 214·37-s − 15·39-s + 360·41-s − 203·43-s − 78·47-s − 174·49-s + 234·51-s − 636·53-s + 195·57-s + 786·59-s + 467·61-s + 117·63-s + 217·67-s − 414·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.701·7-s + 1/3·9-s + 0.164·11-s − 0.106·13-s + 1.11·17-s + 0.784·19-s + 0.405·21-s − 1.25·23-s + 0.192·27-s + 0.422·29-s + 1.73·31-s + 0.0949·33-s + 0.950·37-s − 0.0615·39-s + 1.37·41-s − 0.719·43-s − 0.242·47-s − 0.507·49-s + 0.642·51-s − 1.64·53-s + 0.453·57-s + 1.73·59-s + 0.980·61-s + 0.233·63-s + 0.395·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.566510023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566510023\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 65 T + p^{3} T^{2} \) |
| 23 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 299 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 + 203 T + p^{3} T^{2} \) |
| 47 | \( 1 + 78 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 786 T + p^{3} T^{2} \) |
| 61 | \( 1 - 467 T + p^{3} T^{2} \) |
| 67 | \( 1 - 217 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 286 T + p^{3} T^{2} \) |
| 79 | \( 1 - 272 T + p^{3} T^{2} \) |
| 83 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 511 T + p^{3} T^{2} \) |
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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
−11.44534146007337712325522862185, −10.16528019223979051809864525326, −9.529437132126977732644354542580, −8.200140493286666703182375449450, −7.78578448778905919675940282436, −6.42388950609142170553637069117, −5.17221051177041945565985999756, −4.00933457553956148010146370143, −2.69118882015012861156818223027, −1.21380612214506098606079871789,
1.21380612214506098606079871789, 2.69118882015012861156818223027, 4.00933457553956148010146370143, 5.17221051177041945565985999756, 6.42388950609142170553637069117, 7.78578448778905919675940282436, 8.200140493286666703182375449450, 9.529437132126977732644354542580, 10.16528019223979051809864525326, 11.44534146007337712325522862185
