| L(s) = 1 | + 2·5-s − 9-s − 4·11-s + 12·19-s − 25-s − 12·29-s + 20·31-s − 12·41-s − 2·45-s − 49-s − 8·55-s + 24·59-s + 28·61-s + 4·71-s + 16·79-s + 81-s − 12·89-s + 24·95-s + 4·99-s − 12·101-s + 28·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s − 1.20·11-s + 2.75·19-s − 1/5·25-s − 2.22·29-s + 3.59·31-s − 1.87·41-s − 0.298·45-s − 1/7·49-s − 1.07·55-s + 3.12·59-s + 3.58·61-s + 0.474·71-s + 1.80·79-s + 1/9·81-s − 1.27·89-s + 2.46·95-s + 0.402·99-s − 1.19·101-s + 2.68·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.436515552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436515552\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.530681493647580361261043034161, −8.463480081959772976007979973459, −8.138637811203088928535621409376, −7.78224271891137787612796157022, −7.17817472778533811535081075798, −7.04974329124393964443565938055, −6.60973848332088299689532317438, −6.12179750475866561004483171454, −5.55660039696811541272041611687, −5.40097187658436340098874843807, −5.23127316502946779644132125279, −4.80914988616471396835310900793, −4.05937509446663104239091134499, −3.68941056449865305864174554438, −3.08034955719380830014662360785, −2.90208718495021697407384421140, −2.14390266808538607461283957311, −2.03282432851473089476239779270, −1.01718630096362437117326236304, −0.66388377888236894178438212334,
0.66388377888236894178438212334, 1.01718630096362437117326236304, 2.03282432851473089476239779270, 2.14390266808538607461283957311, 2.90208718495021697407384421140, 3.08034955719380830014662360785, 3.68941056449865305864174554438, 4.05937509446663104239091134499, 4.80914988616471396835310900793, 5.23127316502946779644132125279, 5.40097187658436340098874843807, 5.55660039696811541272041611687, 6.12179750475866561004483171454, 6.60973848332088299689532317438, 7.04974329124393964443565938055, 7.17817472778533811535081075798, 7.78224271891137787612796157022, 8.138637811203088928535621409376, 8.463480081959772976007979973459, 8.530681493647580361261043034161
