| L(s) = 1 | − 4·7-s − 9-s − 8·11-s − 8·23-s + 2·25-s − 16·37-s − 4·43-s + 9·49-s − 8·53-s + 4·63-s − 4·67-s − 8·71-s + 32·77-s − 16·79-s + 81-s + 8·99-s + 16·107-s − 8·109-s − 12·113-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/3·9-s − 2.41·11-s − 1.66·23-s + 2/5·25-s − 2.63·37-s − 0.609·43-s + 9/7·49-s − 1.09·53-s + 0.503·63-s − 0.488·67-s − 0.949·71-s + 3.64·77-s − 1.80·79-s + 1/9·81-s + 0.804·99-s + 1.54·107-s − 0.766·109-s − 1.12·113-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−7.75501570516351375692546096085, −7.26270660478807336723800636245, −6.94871760617873840590969224743, −6.27186112396098422345080236272, −6.02023576977933694650027183368, −5.43193874572251740212702264033, −5.16145384190585602878038038713, −4.57424827680132262599444575314, −3.83662558163497879652140682817, −3.32090996037263072148751830844, −2.90236419855846527713978462519, −2.42778180511912379025374885803, −1.67879112636996363919353009866, 0, 0,
1.67879112636996363919353009866, 2.42778180511912379025374885803, 2.90236419855846527713978462519, 3.32090996037263072148751830844, 3.83662558163497879652140682817, 4.57424827680132262599444575314, 5.16145384190585602878038038713, 5.43193874572251740212702264033, 6.02023576977933694650027183368, 6.27186112396098422345080236272, 6.94871760617873840590969224743, 7.26270660478807336723800636245, 7.75501570516351375692546096085
