| L(s) = 1 | − 3-s + 7-s − 2·9-s − 2·13-s − 6·17-s − 7·19-s − 21-s + 3·23-s − 5·25-s + 5·27-s − 29-s + 4·31-s + 2·37-s + 2·39-s − 9·41-s − 4·43-s + 9·47-s + 49-s + 6·51-s + 9·53-s + 7·57-s + 6·59-s − 2·61-s − 2·63-s + 13·67-s − 3·69-s + 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.554·13-s − 1.45·17-s − 1.60·19-s − 0.218·21-s + 0.625·23-s − 25-s + 0.962·27-s − 0.185·29-s + 0.718·31-s + 0.328·37-s + 0.320·39-s − 1.40·41-s − 0.609·43-s + 1.31·47-s + 1/7·49-s + 0.840·51-s + 1.23·53-s + 0.927·57-s + 0.781·59-s − 0.256·61-s − 0.251·63-s + 1.58·67-s − 0.361·69-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5985121545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5985121545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.51525988869370, −11.79912102579917, −11.53803468604778, −11.20144120124908, −10.71753104305819, −10.27797537781323, −9.873525980251672, −9.216808563600242, −8.665921929376342, −8.473838027583615, −8.044446879806716, −7.271167521971922, −6.791367482354918, −6.571822780969507, −5.906798397801825, −5.505927037743880, −4.997840850923417, −4.466043752271162, −4.131233500396952, −3.483722819631537, −2.664643584239997, −2.294965061671894, −1.862561875506650, −0.9261529764466067, −0.2344304402506960,
0.2344304402506960, 0.9261529764466067, 1.862561875506650, 2.294965061671894, 2.664643584239997, 3.483722819631537, 4.131233500396952, 4.466043752271162, 4.997840850923417, 5.505927037743880, 5.906798397801825, 6.571822780969507, 6.791367482354918, 7.271167521971922, 8.044446879806716, 8.473838027583615, 8.665921929376342, 9.216808563600242, 9.873525980251672, 10.27797537781323, 10.71753104305819, 11.20144120124908, 11.53803468604778, 11.79912102579917, 12.51525988869370
