| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 4·8-s + 3·9-s + 4·10-s − 6·11-s − 6·12-s − 4·15-s + 5·16-s + 6·18-s + 4·19-s + 6·20-s − 12·22-s − 2·23-s − 8·24-s − 7·25-s − 4·27-s − 10·29-s − 8·30-s + 2·31-s + 6·32-s + 12·33-s + 9·36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s − 1.80·11-s − 1.73·12-s − 1.03·15-s + 5/4·16-s + 1.41·18-s + 0.917·19-s + 1.34·20-s − 2.55·22-s − 0.417·23-s − 1.63·24-s − 7/5·25-s − 0.769·27-s − 1.85·29-s − 1.46·30-s + 0.359·31-s + 1.06·32-s + 2.08·33-s + 3/2·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 65 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 123 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 125 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 202 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 157 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 217 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61023058811030619173799805256, −7.40171622104415464854761343331, −6.90713762571869171625312758424, −6.62669774212253656586969418802, −6.16098280789704842267215232592, −5.96501100391032567632797220672, −5.44678503670885078035586986734, −5.41628733234915163876037836263, −5.04720565252523396300272401375, −4.92525070376229147308332762605, −4.08864332762431854794004770252, −4.04104206316214651592095362413, −3.25848786985001685933684158774, −3.25490929349754338752633539533, −2.33081473927780838828836060826, −2.30885965474843805526523537929, −1.48722583567272798780258746032, −1.47442069596337445812573067496, 0, 0,
1.47442069596337445812573067496, 1.48722583567272798780258746032, 2.30885965474843805526523537929, 2.33081473927780838828836060826, 3.25490929349754338752633539533, 3.25848786985001685933684158774, 4.04104206316214651592095362413, 4.08864332762431854794004770252, 4.92525070376229147308332762605, 5.04720565252523396300272401375, 5.41628733234915163876037836263, 5.44678503670885078035586986734, 5.96501100391032567632797220672, 6.16098280789704842267215232592, 6.62669774212253656586969418802, 6.90713762571869171625312758424, 7.40171622104415464854761343331, 7.61023058811030619173799805256
