L(s) = 1 | + 3-s + 4-s + 8·7-s + 9-s + 12-s − 2·13-s + 16-s + 8·19-s + 8·21-s + 25-s + 27-s + 8·28-s − 16·31-s + 36-s + 4·37-s − 2·39-s + 24·43-s + 48-s + 34·49-s − 2·52-s + 8·57-s − 20·61-s + 8·63-s + 64-s − 8·67-s − 12·73-s + 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/2·4-s + 3.02·7-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 1.83·19-s + 1.74·21-s + 1/5·25-s + 0.192·27-s + 1.51·28-s − 2.87·31-s + 1/6·36-s + 0.657·37-s − 0.320·39-s + 3.65·43-s + 0.144·48-s + 34/7·49-s − 0.277·52-s + 1.05·57-s − 2.56·61-s + 1.00·63-s + 1/8·64-s − 0.977·67-s − 1.40·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.213078166\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.213078166\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.642487975353242862587712517060, −7.70002191137841568751282880154, −7.61942621520258683571159755696, −7.61499574063730421950670590635, −7.15844162242666446745691958741, −6.12276299001965025887943314435, −5.47980176854220799955648347826, −5.42080963650772912060550607812, −4.68667184047043357988258422645, −4.33965496590448459865286945078, −3.75778333243500449289895503228, −2.83072987113389395332805085297, −2.40122970806064783978264870635, −1.46269795593787510909097790861, −1.40608557557707214069713458551,
1.40608557557707214069713458551, 1.46269795593787510909097790861, 2.40122970806064783978264870635, 2.83072987113389395332805085297, 3.75778333243500449289895503228, 4.33965496590448459865286945078, 4.68667184047043357988258422645, 5.42080963650772912060550607812, 5.47980176854220799955648347826, 6.12276299001965025887943314435, 7.15844162242666446745691958741, 7.61499574063730421950670590635, 7.61942621520258683571159755696, 7.70002191137841568751282880154, 8.642487975353242862587712517060