| L(s) = 1 | + 4·3-s − 2·5-s + 6·9-s − 4·13-s − 8·15-s − 25-s − 4·27-s − 20·37-s − 16·39-s − 12·41-s + 12·43-s − 12·45-s + 2·49-s + 12·53-s + 8·65-s + 20·67-s − 24·71-s − 4·75-s + 16·79-s − 37·81-s − 12·83-s − 4·89-s − 4·107-s − 80·111-s − 24·117-s − 18·121-s − 48·123-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 2·9-s − 1.10·13-s − 2.06·15-s − 1/5·25-s − 0.769·27-s − 3.28·37-s − 2.56·39-s − 1.87·41-s + 1.82·43-s − 1.78·45-s + 2/7·49-s + 1.64·53-s + 0.992·65-s + 2.44·67-s − 2.84·71-s − 0.461·75-s + 1.80·79-s − 4.11·81-s − 1.31·83-s − 0.423·89-s − 0.386·107-s − 7.59·111-s − 2.21·117-s − 1.63·121-s − 4.32·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−8.499715772662989764702573299468, −8.098681382024412731658064042949, −7.48649464154346783477619856196, −7.33738117792255002879146989083, −6.95023759239239106595773853946, −6.15302264435441050088112742528, −5.26498006511685590910060560318, −5.16542238567969836254694300670, −4.04799601522778258065319452547, −3.91058908136761919842529082491, −3.35101374838703231470604183982, −2.81830188953599087445367836313, −2.30676446591740616355718333113, −1.71401499382330216031350685870, 0,
1.71401499382330216031350685870, 2.30676446591740616355718333113, 2.81830188953599087445367836313, 3.35101374838703231470604183982, 3.91058908136761919842529082491, 4.04799601522778258065319452547, 5.16542238567969836254694300670, 5.26498006511685590910060560318, 6.15302264435441050088112742528, 6.95023759239239106595773853946, 7.33738117792255002879146989083, 7.48649464154346783477619856196, 8.098681382024412731658064042949, 8.499715772662989764702573299468
