L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·11-s − 14-s + 16-s − 2·17-s + 6·19-s + 2·22-s − 6·23-s + 28-s − 3·29-s + 5·31-s − 32-s + 2·34-s − 5·37-s − 6·38-s + 5·41-s − 4·43-s − 2·44-s + 6·46-s + 47-s + 49-s + 2·53-s − 56-s + 3·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.426·22-s − 1.25·23-s + 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s + 0.342·34-s − 0.821·37-s − 0.973·38-s + 0.780·41-s − 0.609·43-s − 0.301·44-s + 0.884·46-s + 0.145·47-s + 1/7·49-s + 0.274·53-s − 0.133·56-s + 0.393·58-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.49316919103427156614328538070, −6.88807175918362071558871138142, −5.98255577252425162544416451881, −5.43890028656806183276435226430, −4.62085660215336373981708478807, −3.74655847589742477364805355968, −2.85043699210536436087673694696, −2.08584932175036929894509598293, −1.16996431546132010950924371010, 0,
1.16996431546132010950924371010, 2.08584932175036929894509598293, 2.85043699210536436087673694696, 3.74655847589742477364805355968, 4.62085660215336373981708478807, 5.43890028656806183276435226430, 5.98255577252425162544416451881, 6.88807175918362071558871138142, 7.49316919103427156614328538070