| L(s) = 1 | − 3-s − 2·9-s + 3·11-s − 6·19-s + 25-s + 5·27-s − 3·33-s − 3·41-s + 9·43-s + 49-s + 6·57-s + 3·59-s − 3·67-s + 2·73-s − 75-s + 81-s − 18·83-s − 11·97-s − 6·99-s − 12·107-s − 13·121-s + 3·123-s + 127-s − 9·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.904·11-s − 1.37·19-s + 1/5·25-s + 0.962·27-s − 0.522·33-s − 0.468·41-s + 1.37·43-s + 1/7·49-s + 0.794·57-s + 0.390·59-s − 0.366·67-s + 0.234·73-s − 0.115·75-s + 1/9·81-s − 1.97·83-s − 1.11·97-s − 0.603·99-s − 1.16·107-s − 1.18·121-s + 0.270·123-s + 0.0887·127-s − 0.792·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.80281209697826320614665815067, −7.14688163685371039933020265366, −6.82779965554688579966104708540, −6.45447236289726831616702395892, −5.98202692106423447732455070940, −5.64286290070652482374673104137, −5.18906004654302926177572836260, −4.51057351572907322944852351942, −4.21642580338116036251398749230, −3.70854541531314082311497030642, −2.98765438375415752686837114256, −2.51152965737044000816090701067, −1.78889623328278788039295484359, −0.997446232886411073455495177851, 0,
0.997446232886411073455495177851, 1.78889623328278788039295484359, 2.51152965737044000816090701067, 2.98765438375415752686837114256, 3.70854541531314082311497030642, 4.21642580338116036251398749230, 4.51057351572907322944852351942, 5.18906004654302926177572836260, 5.64286290070652482374673104137, 5.98202692106423447732455070940, 6.45447236289726831616702395892, 6.82779965554688579966104708540, 7.14688163685371039933020265366, 7.80281209697826320614665815067
