| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 5·11-s + 16-s + 17-s + 18-s − 6·19-s − 5·22-s − 8·25-s + 32-s + 34-s + 36-s − 6·38-s + 2·41-s + 13·43-s − 5·44-s − 10·49-s − 8·50-s − 59-s + 64-s + 12·67-s + 68-s + 72-s − 73-s − 6·76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s − 1.50·11-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s − 1.06·22-s − 8/5·25-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.973·38-s + 0.312·41-s + 1.98·43-s − 0.753·44-s − 1.42·49-s − 1.13·50-s − 0.130·59-s + 1/8·64-s + 1.46·67-s + 0.121·68-s + 0.117·72-s − 0.117·73-s − 0.688·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 436608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 436608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 379 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.148650602089301963613882676527, −7.928194216952050789093547040861, −7.48808007102418684033128033431, −6.96151373883176490069848760100, −6.43793632451480372069369828847, −5.91309245421776711486562902768, −5.57240441987483691786781988995, −5.05347241573623230304646412055, −4.45439572222945557803158085175, −4.04398806016294242476820602395, −3.50891187704449823112941599299, −2.56773828322038018309412400710, −2.40607160438023610973650559060, −1.46892909049898633307261346484, 0,
1.46892909049898633307261346484, 2.40607160438023610973650559060, 2.56773828322038018309412400710, 3.50891187704449823112941599299, 4.04398806016294242476820602395, 4.45439572222945557803158085175, 5.05347241573623230304646412055, 5.57240441987483691786781988995, 5.91309245421776711486562902768, 6.43793632451480372069369828847, 6.96151373883176490069848760100, 7.48808007102418684033128033431, 7.928194216952050789093547040861, 8.148650602089301963613882676527
