| L(s) = 1 | − 7·3-s + 7·5-s − 28·7-s + 27·9-s + 5·11-s + 28·13-s − 49·15-s + 21·17-s − 49·19-s + 196·21-s − 159·23-s + 125·25-s − 224·27-s − 116·29-s + 147·31-s − 35·33-s − 196·35-s + 219·37-s − 196·39-s + 700·41-s − 248·43-s + 189·45-s + 525·47-s + 441·49-s − 147·51-s + 303·53-s + 35·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.626·5-s − 1.51·7-s + 9-s + 0.137·11-s + 0.597·13-s − 0.843·15-s + 0.299·17-s − 0.591·19-s + 2.03·21-s − 1.44·23-s + 25-s − 1.59·27-s − 0.742·29-s + 0.851·31-s − 0.184·33-s − 0.946·35-s + 0.973·37-s − 0.804·39-s + 2.66·41-s − 0.879·43-s + 0.626·45-s + 1.62·47-s + 9/7·49-s − 0.403·51-s + 0.785·53-s + 0.0858·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.125126307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125126307\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T - 76 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T - 1306 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 21 T - 4472 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 49 T - 4458 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 159 T + 13114 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 147 T - 8182 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 219 T - 2692 T^{2} - 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 350 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 525 T + 171802 T^{2} - 525 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 303 T - 57068 T^{2} - 303 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 105 T - 194354 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 413 T - 56412 T^{2} + 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 415 T - 128538 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1113 T + 849752 T^{2} - 1113 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 103 T - 482430 T^{2} + 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1092 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 329 T - 596728 T^{2} - 329 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 882 T + p^{3} T^{2} )^{2} \) |
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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
−10.87855945724156514988503179153, −10.62240506139489001272366421856, −9.927167967635306365509112811152, −9.713611102826114979726388171354, −9.238251935205263441509814225402, −8.931151580329590855320822137884, −7.88513942124747585222291616700, −7.83310283940166509979952667173, −6.88895420754431933082961305050, −6.58506500854899234003332770020, −6.02190645106072940859777272613, −5.94806035131416538640983497923, −5.45420908597154314011653139094, −4.69581635357626105327492648608, −3.91933302504010352607203252584, −3.75264328317960321226236103452, −2.67423363470333866436314425923, −2.15094185677700939001104097232, −1.03785500779598773030214159615, −0.44827821665166528570442649629,
0.44827821665166528570442649629, 1.03785500779598773030214159615, 2.15094185677700939001104097232, 2.67423363470333866436314425923, 3.75264328317960321226236103452, 3.91933302504010352607203252584, 4.69581635357626105327492648608, 5.45420908597154314011653139094, 5.94806035131416538640983497923, 6.02190645106072940859777272613, 6.58506500854899234003332770020, 6.88895420754431933082961305050, 7.83310283940166509979952667173, 7.88513942124747585222291616700, 8.931151580329590855320822137884, 9.238251935205263441509814225402, 9.713611102826114979726388171354, 9.927167967635306365509112811152, 10.62240506139489001272366421856, 10.87855945724156514988503179153
