| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 3·11-s + 4·13-s + 14-s + 16-s + 17-s − 5·19-s − 3·22-s + 8·23-s − 4·26-s − 28-s + 4·29-s − 3·31-s − 32-s − 34-s + 7·37-s + 5·38-s − 2·41-s + 43-s + 3·44-s − 8·46-s − 7·47-s − 6·49-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.639·22-s + 1.66·23-s − 0.784·26-s − 0.188·28-s + 0.742·29-s − 0.538·31-s − 0.176·32-s − 0.171·34-s + 1.15·37-s + 0.811·38-s − 0.312·41-s + 0.152·43-s + 0.452·44-s − 1.17·46-s − 1.02·47-s − 6/7·49-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.591146593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591146593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.135292844046089664559906175869, −7.01740364334808750521314711647, −6.63898337253409794146618749411, −6.05399666296688153162808443071, −5.14787043994372317388094418023, −4.15843427047876882349228250562, −3.47462301731735252674209775046, −2.62891927239608265743979570517, −1.54999844370951157077561065508, −0.74929188852310592680180496602,
0.74929188852310592680180496602, 1.54999844370951157077561065508, 2.62891927239608265743979570517, 3.47462301731735252674209775046, 4.15843427047876882349228250562, 5.14787043994372317388094418023, 6.05399666296688153162808443071, 6.63898337253409794146618749411, 7.01740364334808750521314711647, 8.135292844046089664559906175869
