| L(s) = 1 | + 2-s − 5-s − 5·7-s − 8-s − 10-s − 11-s + 4·13-s − 5·14-s − 16-s + 5·17-s + 6·19-s − 22-s + 7·23-s + 5·25-s + 4·26-s + 16·29-s − 10·31-s + 5·34-s + 5·35-s + 8·37-s + 6·38-s + 40-s − 14·41-s + 8·43-s + 7·46-s − 47-s + 18·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.213·22-s + 1.45·23-s + 25-s + 0.784·26-s + 2.97·29-s − 1.79·31-s + 0.857·34-s + 0.845·35-s + 1.31·37-s + 0.973·38-s + 0.158·40-s − 2.18·41-s + 1.21·43-s + 1.03·46-s − 0.145·47-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.773316029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.773316029\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−9.722942406712210365253753669600, −9.384282573028651789534262960905, −8.912745131488240073196171624940, −8.803437196083605841855088858414, −8.026861769763379612240257243805, −7.83387624338365706137608638082, −7.16336257771782792552786202175, −6.78158456778465121390164540720, −6.55421423571550900349679512308, −6.10146751378147095158105901835, −5.50847474699024380613983410841, −5.21766004599126879123267303568, −4.84087452066317297379660552526, −4.05529041732218200274643882425, −3.64370442573907985162946833372, −3.21729307601242294853268920414, −3.03836291788202732639667413891, −2.46910547395423037472204259919, −1.07291782668336743015093228662, −0.77161620851336411024148920747,
0.77161620851336411024148920747, 1.07291782668336743015093228662, 2.46910547395423037472204259919, 3.03836291788202732639667413891, 3.21729307601242294853268920414, 3.64370442573907985162946833372, 4.05529041732218200274643882425, 4.84087452066317297379660552526, 5.21766004599126879123267303568, 5.50847474699024380613983410841, 6.10146751378147095158105901835, 6.55421423571550900349679512308, 6.78158456778465121390164540720, 7.16336257771782792552786202175, 7.83387624338365706137608638082, 8.026861769763379612240257243805, 8.803437196083605841855088858414, 8.912745131488240073196171624940, 9.384282573028651789534262960905, 9.722942406712210365253753669600
