| L(s) = 1 | − 3·3-s + 4-s + 3·5-s + 5·7-s + 6·9-s − 3·12-s − 9·15-s − 3·16-s + 3·20-s − 15·21-s − 3·25-s − 9·27-s + 5·28-s + 3·29-s + 31-s + 15·35-s + 6·36-s + 10·43-s + 18·45-s + 9·48-s + 7·49-s − 9·60-s + 7·61-s + 30·63-s − 7·64-s + 9·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s + 1.34·5-s + 1.88·7-s + 2·9-s − 0.866·12-s − 2.32·15-s − 3/4·16-s + 0.670·20-s − 3.27·21-s − 3/5·25-s − 1.73·27-s + 0.944·28-s + 0.557·29-s + 0.179·31-s + 2.53·35-s + 36-s + 1.52·43-s + 2.68·45-s + 1.29·48-s + 49-s − 1.16·60-s + 0.896·61-s + 3.77·63-s − 7/8·64-s + 1.06·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.573607906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573607906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 109 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−9.559477245666261431748240340229, −9.276992247539270193564378985661, −8.496972971786191896084010778174, −7.88483568113115738210596155884, −7.49943638232493936284055289999, −6.81037037640430312565013328570, −6.36700557394260242480659872109, −5.90313780420689699354028993445, −5.49057836017641940440672571376, −4.87256908837552629042188339614, −4.66181032046370089202988240046, −3.82940333627671916288200912699, −2.32751750043690749131518887152, −1.94890332842232973277070078772, −1.09949188504807237006178830172,
1.09949188504807237006178830172, 1.94890332842232973277070078772, 2.32751750043690749131518887152, 3.82940333627671916288200912699, 4.66181032046370089202988240046, 4.87256908837552629042188339614, 5.49057836017641940440672571376, 5.90313780420689699354028993445, 6.36700557394260242480659872109, 6.81037037640430312565013328570, 7.49943638232493936284055289999, 7.88483568113115738210596155884, 8.496972971786191896084010778174, 9.276992247539270193564378985661, 9.559477245666261431748240340229
