| L(s) = 1 | + 2-s − 2·7-s − 8-s + 11-s + 4·13-s − 2·14-s − 16-s − 12·17-s − 8·19-s + 22-s − 6·23-s + 5·25-s + 4·26-s − 6·29-s − 8·31-s − 12·34-s − 20·37-s − 8·38-s − 6·41-s − 8·43-s − 6·46-s + 6·47-s + 7·49-s + 5·50-s + 2·56-s − 6·58-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.755·7-s − 0.353·8-s + 0.301·11-s + 1.10·13-s − 0.534·14-s − 1/4·16-s − 2.91·17-s − 1.83·19-s + 0.213·22-s − 1.25·23-s + 25-s + 0.784·26-s − 1.11·29-s − 1.43·31-s − 2.05·34-s − 3.28·37-s − 1.29·38-s − 0.937·41-s − 1.21·43-s − 0.884·46-s + 0.875·47-s + 49-s + 0.707·50-s + 0.267·56-s − 0.787·58-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.823522586597613641969185713155, −8.796616134912811635432916400572, −8.465127065966630089161874736856, −8.156049874949185859726919672775, −7.17818355183004524189067535742, −6.86435930265616229656004053785, −6.70364137603408796901837619544, −6.42119987478575249054089169467, −5.78212363186818498779572422052, −5.52010695050081761890447216609, −4.94699855274743047318720614922, −4.42693962824454371013937712665, −3.93932028210727303210167668542, −3.83457998402915164076033113796, −3.30630139689748606690655339026, −2.56830513159392114060902556043, −1.84607312053401710187327853531, −1.81584022054677731782935524934, 0, 0,
1.81584022054677731782935524934, 1.84607312053401710187327853531, 2.56830513159392114060902556043, 3.30630139689748606690655339026, 3.83457998402915164076033113796, 3.93932028210727303210167668542, 4.42693962824454371013937712665, 4.94699855274743047318720614922, 5.52010695050081761890447216609, 5.78212363186818498779572422052, 6.42119987478575249054089169467, 6.70364137603408796901837619544, 6.86435930265616229656004053785, 7.17818355183004524189067535742, 8.156049874949185859726919672775, 8.465127065966630089161874736856, 8.796616134912811635432916400572, 8.823522586597613641969185713155
