| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 2·9-s − 2·10-s + 2·12-s − 3·13-s + 15-s − 4·16-s + 4·18-s + 2·20-s + 25-s + 6·26-s − 5·27-s − 2·30-s − 2·31-s + 8·32-s − 4·36-s + 12·37-s − 3·39-s − 3·41-s − 3·43-s − 2·45-s − 4·48-s − 10·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 2/3·9-s − 0.632·10-s + 0.577·12-s − 0.832·13-s + 0.258·15-s − 16-s + 0.942·18-s + 0.447·20-s + 1/5·25-s + 1.17·26-s − 0.962·27-s − 0.365·30-s − 0.359·31-s + 1.41·32-s − 2/3·36-s + 1.97·37-s − 0.480·39-s − 0.468·41-s − 0.457·43-s − 0.298·45-s − 0.577·48-s − 1.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 176 T^{2} + p^{2} T^{4} \) |
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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
−8.719478234935615867978748867412, −8.478896065053912585392389292237, −8.060112442021522775444254290557, −7.37098860384523408563181956512, −7.28707590353294566879593311442, −6.56325927203695683308289102646, −5.92677060926460013974694426257, −5.55403496446035493943234474244, −4.67832390952108040223362382520, −4.33711019032836985145843407755, −3.30018857754011546651837966620, −2.72648121336042217986362913729, −2.15117827428324848537758894218, −1.34441859288597537687459084223, 0,
1.34441859288597537687459084223, 2.15117827428324848537758894218, 2.72648121336042217986362913729, 3.30018857754011546651837966620, 4.33711019032836985145843407755, 4.67832390952108040223362382520, 5.55403496446035493943234474244, 5.92677060926460013974694426257, 6.56325927203695683308289102646, 7.28707590353294566879593311442, 7.37098860384523408563181956512, 8.060112442021522775444254290557, 8.478896065053912585392389292237, 8.719478234935615867978748867412
