| L(s) = 1 | + 2-s − 4-s − 5-s − 7-s − 3·8-s − 10-s + 2·13-s − 14-s − 16-s + 6·17-s − 4·19-s + 20-s + 23-s + 25-s + 2·26-s + 28-s + 2·29-s − 4·31-s + 5·32-s + 6·34-s + 35-s − 6·37-s − 4·38-s + 3·40-s − 2·41-s + 4·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s − 0.718·31-s + 0.883·32-s + 1.02·34-s + 0.169·35-s − 0.986·37-s − 0.648·38-s + 0.474·40-s − 0.312·41-s + 0.609·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.56996413114312560916596829180, −6.72937945027526351263770390922, −6.02870849459207800684675372358, −5.39538957675051925123528952195, −4.71554370886427380270606334475, −3.79101281542018080394874707436, −3.50726525375490079860283142191, −2.56019334354623730417867222947, −1.18548187973464614514223978526, 0,
1.18548187973464614514223978526, 2.56019334354623730417867222947, 3.50726525375490079860283142191, 3.79101281542018080394874707436, 4.71554370886427380270606334475, 5.39538957675051925123528952195, 6.02870849459207800684675372358, 6.72937945027526351263770390922, 7.56996413114312560916596829180
