| L(s) = 1 | + 2·2-s + 2·4-s − 9-s − 2·11-s − 4·16-s − 2·18-s − 4·22-s + 4·23-s + 25-s − 6·29-s − 8·32-s − 2·36-s + 4·37-s + 8·43-s − 4·44-s + 8·46-s + 2·50-s − 4·53-s − 12·58-s − 8·64-s + 4·67-s − 24·71-s + 8·74-s − 2·79-s − 8·81-s + 16·86-s + 8·92-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1/3·9-s − 0.603·11-s − 16-s − 0.471·18-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.41·32-s − 1/3·36-s + 0.657·37-s + 1.21·43-s − 0.603·44-s + 1.17·46-s + 0.282·50-s − 0.549·53-s − 1.57·58-s − 64-s + 0.488·67-s − 2.84·71-s + 0.929·74-s − 0.225·79-s − 8/9·81-s + 1.72·86-s + 0.834·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
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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
−7.68117196100858044910854197940, −7.46853029772026199881711626115, −6.94324729295070658022845854153, −6.44682343500939274218713527376, −5.93053499443198491651721673028, −5.66826214064922511970416222665, −5.16875667028351442657090636757, −4.77447533845142716323907996027, −4.20985376594610484840421053346, −3.84383350810850285981315496000, −3.11976846234536684559706968177, −2.78910588038807023546012165014, −2.25172209160790035948633686812, −1.31591766762650749547578441743, 0,
1.31591766762650749547578441743, 2.25172209160790035948633686812, 2.78910588038807023546012165014, 3.11976846234536684559706968177, 3.84383350810850285981315496000, 4.20985376594610484840421053346, 4.77447533845142716323907996027, 5.16875667028351442657090636757, 5.66826214064922511970416222665, 5.93053499443198491651721673028, 6.44682343500939274218713527376, 6.94324729295070658022845854153, 7.46853029772026199881711626115, 7.68117196100858044910854197940
