| L(s) = 1 | + 2·5-s + 4·7-s + 2·9-s + 8·19-s + 3·25-s + 8·35-s − 4·37-s + 4·43-s + 4·45-s + 2·49-s + 12·53-s + 8·63-s − 5·81-s − 12·83-s + 4·89-s + 16·95-s − 12·97-s − 4·107-s − 28·113-s − 11·121-s + 4·125-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s + 2/3·9-s + 1.83·19-s + 3/5·25-s + 1.35·35-s − 0.657·37-s + 0.609·43-s + 0.596·45-s + 2/7·49-s + 1.64·53-s + 1.00·63-s − 5/9·81-s − 1.31·83-s + 0.423·89-s + 1.64·95-s − 1.21·97-s − 0.386·107-s − 2.63·113-s − 121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.159897137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.159897137\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.618682628665420280996220498022, −8.240222389730593457964628653362, −7.70200207293889953687826754199, −7.32391724627428462753174682459, −6.93982858233685330930908381046, −6.33583797650597465447621281920, −5.67024949903968962367204137625, −5.24994886271652254981768200541, −5.06084578965330143300269549972, −4.31158463954656220302583301841, −3.85309381603312166996610685441, −2.98808210810482496056029083883, −2.37921741827803489995585398947, −1.54617278880287985105468381852, −1.18789081324829946125149839911,
1.18789081324829946125149839911, 1.54617278880287985105468381852, 2.37921741827803489995585398947, 2.98808210810482496056029083883, 3.85309381603312166996610685441, 4.31158463954656220302583301841, 5.06084578965330143300269549972, 5.24994886271652254981768200541, 5.67024949903968962367204137625, 6.33583797650597465447621281920, 6.93982858233685330930908381046, 7.32391724627428462753174682459, 7.70200207293889953687826754199, 8.240222389730593457964628653362, 8.618682628665420280996220498022
