| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 5·11-s + 14-s + 16-s + 2·17-s − 19-s − 5·22-s − 23-s + 28-s − 4·29-s − 9·31-s + 32-s + 2·34-s − 5·37-s − 38-s + 9·41-s + 10·43-s − 5·44-s − 46-s + 6·47-s + 49-s + 12·53-s + 56-s − 4·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.50·11-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 1.06·22-s − 0.208·23-s + 0.188·28-s − 0.742·29-s − 1.61·31-s + 0.176·32-s + 0.342·34-s − 0.821·37-s − 0.162·38-s + 1.40·41-s + 1.52·43-s − 0.753·44-s − 0.147·46-s + 0.875·47-s + 1/7·49-s + 1.64·53-s + 0.133·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.013104356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013104356\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.46782089600524145924378084994, −7.23121471355869503316580940451, −6.11146364437453685304727079614, −5.38728005132442870577236549443, −5.22933389027289580544317892179, −4.10619494891641938149588170500, −3.61821594379271561802584596428, −2.50798084154612067735059981675, −2.09937771523637851330725269426, −0.72299960170725188223213743879,
0.72299960170725188223213743879, 2.09937771523637851330725269426, 2.50798084154612067735059981675, 3.61821594379271561802584596428, 4.10619494891641938149588170500, 5.22933389027289580544317892179, 5.38728005132442870577236549443, 6.11146364437453685304727079614, 7.23121471355869503316580940451, 7.46782089600524145924378084994
