| 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 − 2·11-s − 6·12-s − 4·13-s + 4·15-s + 5·16-s + 2·17-s + 6·18-s − 6·20-s − 4·22-s − 8·24-s + 3·25-s − 8·26-s − 4·27-s − 4·29-s + 8·30-s + 6·32-s + 4·33-s + 4·34-s + 9·36-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 − 0.603·11-s − 1.73·12-s − 1.10·13-s + 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 1.34·20-s − 0.852·22-s − 1.63·24-s + 3/5·25-s − 1.56·26-s − 0.769·27-s − 0.742·29-s + 1.46·30-s + 1.06·32-s + 0.696·33-s + 0.685·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.78612519108007866032157127294, −7.61999675736610248573624129939, −6.97682742576427042492025123265, −6.94459757511482332578498549992, −6.34175716904771090287401124688, −6.19878126485364920500423828762, −5.68138842777879928251689868997, −5.35368982806586943985244580870, −5.06413793688850314468530698161, −4.63818501502965074946873381586, −4.42178074759911517124284342424, −4.13561012452683558319650626772, −3.40686618502901723434424123273, −3.30529727003089786545209834148, −2.54058200587719969399625867769, −2.50211085707945536183563892025, −1.37926337551200595808324804532, −1.37098149427847210554442280434, 0, 0,
1.37098149427847210554442280434, 1.37926337551200595808324804532, 2.50211085707945536183563892025, 2.54058200587719969399625867769, 3.30529727003089786545209834148, 3.40686618502901723434424123273, 4.13561012452683558319650626772, 4.42178074759911517124284342424, 4.63818501502965074946873381586, 5.06413793688850314468530698161, 5.35368982806586943985244580870, 5.68138842777879928251689868997, 6.19878126485364920500423828762, 6.34175716904771090287401124688, 6.94459757511482332578498549992, 6.97682742576427042492025123265, 7.61999675736610248573624129939, 7.78612519108007866032157127294
