| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 3·9-s − 12-s − 2·13-s + 14-s + 16-s + 3·18-s + 21-s + 24-s + 2·25-s + 2·26-s + 4·27-s − 28-s + 5·31-s − 32-s − 3·36-s + 2·39-s − 42-s − 48-s − 6·49-s − 2·50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 9-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.707·18-s + 0.218·21-s + 0.204·24-s + 2/5·25-s + 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.898·31-s − 0.176·32-s − 1/2·36-s + 0.320·39-s − 0.154·42-s − 0.144·48-s − 6/7·49-s − 0.282·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 31 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−8.441324549305048793243198735581, −8.201206039480013829837561918370, −7.55406681052510894201697470316, −6.88069231238119501762592478149, −6.81111680381649210828792579680, −6.10514497040304941987819606217, −5.69441580329189858879621669958, −5.26591345728831246455262417610, −4.68490438080432213909851845231, −4.00375488849004931426370015424, −3.22514396276328070304497740741, −2.75083861704259195741144047437, −2.11907557010753350787321255304, −1.00016023716685044817082251830, 0,
1.00016023716685044817082251830, 2.11907557010753350787321255304, 2.75083861704259195741144047437, 3.22514396276328070304497740741, 4.00375488849004931426370015424, 4.68490438080432213909851845231, 5.26591345728831246455262417610, 5.69441580329189858879621669958, 6.10514497040304941987819606217, 6.81111680381649210828792579680, 6.88069231238119501762592478149, 7.55406681052510894201697470316, 8.201206039480013829837561918370, 8.441324549305048793243198735581
