| L(s) = 1 | − 2·3-s + 4-s + 7-s + 9-s − 2·12-s + 16-s + 12·19-s − 2·21-s − 2·25-s + 4·27-s + 28-s − 8·31-s + 36-s − 12·37-s − 20·43-s − 2·48-s + 49-s − 24·57-s + 12·61-s + 63-s + 64-s + 8·67-s − 15·73-s + 4·75-s + 12·76-s + 16·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 2.75·19-s − 0.436·21-s − 2/5·25-s + 0.769·27-s + 0.188·28-s − 1.43·31-s + 1/6·36-s − 1.97·37-s − 3.04·43-s − 0.288·48-s + 1/7·49-s − 3.17·57-s + 1.53·61-s + 0.125·63-s + 1/8·64-s + 0.977·67-s − 1.75·73-s + 0.461·75-s + 1.37·76-s + 1.80·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376835233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376835233\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 14 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.104896712313343120997320633635, −7.44591300444563061205382192540, −7.38143668393976292143625351940, −6.80000171759750942302082249743, −6.46297831799875791051495100213, −5.83721704756890568930732991673, −5.36981291871743877546857941859, −5.14030387392133989849927045411, −4.90082812907387706804743696889, −3.89423478851217689693985034568, −3.36141597922135942136294390043, −3.09179969440150036582632462395, −1.99354823677226090726908316526, −1.53607576198637436986990905357, −0.60845760105563453073349543843,
0.60845760105563453073349543843, 1.53607576198637436986990905357, 1.99354823677226090726908316526, 3.09179969440150036582632462395, 3.36141597922135942136294390043, 3.89423478851217689693985034568, 4.90082812907387706804743696889, 5.14030387392133989849927045411, 5.36981291871743877546857941859, 5.83721704756890568930732991673, 6.46297831799875791051495100213, 6.80000171759750942302082249743, 7.38143668393976292143625351940, 7.44591300444563061205382192540, 8.104896712313343120997320633635
