| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 5·11-s + 4·13-s + 3·14-s + 16-s + 17-s − 19-s + 5·22-s + 4·23-s + 4·26-s + 3·28-s + 4·29-s − 31-s + 32-s + 34-s − 9·37-s − 38-s + 10·41-s + 11·43-s + 5·44-s + 4·46-s − 9·47-s + 2·49-s + 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 1.50·11-s + 1.10·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s − 0.229·19-s + 1.06·22-s + 0.834·23-s + 0.784·26-s + 0.566·28-s + 0.742·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s − 1.47·37-s − 0.162·38-s + 1.56·41-s + 1.67·43-s + 0.753·44-s + 0.589·46-s − 1.31·47-s + 2/7·49-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
| \(L(1)\) |
\(\approx\) |
\(4.913132052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.913132052\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 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
−7.72878044023989817034248599610, −7.12943392685029533863993118291, −6.30624121489563225366616238655, −5.85912468415804491111448923105, −4.91632750297226823841529137612, −4.33255169537249309199836097536, −3.69836703723790804402987926743, −2.84212101092100100123149762548, −1.63503664792587817456027608506, −1.16861012176362650589223738464,
1.16861012176362650589223738464, 1.63503664792587817456027608506, 2.84212101092100100123149762548, 3.69836703723790804402987926743, 4.33255169537249309199836097536, 4.91632750297226823841529137612, 5.85912468415804491111448923105, 6.30624121489563225366616238655, 7.12943392685029533863993118291, 7.72878044023989817034248599610
