| L(s) = 1 | + 2·5-s − 10·11-s + 14·19-s − 25-s + 10·29-s − 6·31-s + 14·41-s + 10·49-s − 20·55-s + 30·59-s + 28·61-s − 10·71-s + 16·79-s − 6·89-s + 28·95-s − 30·101-s + 18·109-s + 53·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s − 12·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.01·11-s + 3.21·19-s − 1/5·25-s + 1.85·29-s − 1.07·31-s + 2.18·41-s + 10/7·49-s − 2.69·55-s + 3.90·59-s + 3.58·61-s − 1.18·71-s + 1.80·79-s − 0.635·89-s + 2.87·95-s − 2.98·101-s + 1.72·109-s + 4.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s − 0.963·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.412891768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.412891768\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.743536939271235925691531680220, −8.402061150344854691751697261604, −8.100192861357411231462897940024, −7.68783460011982158259623737201, −7.28757851154520688005287227432, −7.21706996927050284671217656385, −6.68552202568527071563045213833, −5.99845363328289636055293638855, −5.59906970894942676522417154246, −5.38675684065261324655205415692, −5.20287668232690791959495844249, −4.97174177552502927701994610221, −3.99992380927918166722556576673, −3.86091550604446178400907312555, −2.88684673687198602138846424768, −2.87432077235443808160423520517, −2.41147907096045077366422783657, −1.99822011138683783004596458965, −0.881200048699571666827190425284, −0.75444786489756657512384118261,
0.75444786489756657512384118261, 0.881200048699571666827190425284, 1.99822011138683783004596458965, 2.41147907096045077366422783657, 2.87432077235443808160423520517, 2.88684673687198602138846424768, 3.86091550604446178400907312555, 3.99992380927918166722556576673, 4.97174177552502927701994610221, 5.20287668232690791959495844249, 5.38675684065261324655205415692, 5.59906970894942676522417154246, 5.99845363328289636055293638855, 6.68552202568527071563045213833, 7.21706996927050284671217656385, 7.28757851154520688005287227432, 7.68783460011982158259623737201, 8.100192861357411231462897940024, 8.402061150344854691751697261604, 8.743536939271235925691531680220
