| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 2·11-s − 4·15-s + 2·23-s + 3·25-s − 4·27-s + 4·31-s − 4·33-s − 12·37-s − 2·45-s + 6·47-s + 10·49-s + 8·53-s + 4·55-s − 14·67-s + 4·69-s + 20·71-s + 6·75-s − 11·81-s + 4·89-s + 8·93-s + 4·97-s − 2·99-s + 2·103-s − 24·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 1.03·15-s + 0.417·23-s + 3/5·25-s − 0.769·27-s + 0.718·31-s − 0.696·33-s − 1.97·37-s − 0.298·45-s + 0.875·47-s + 10/7·49-s + 1.09·53-s + 0.539·55-s − 1.71·67-s + 0.481·69-s + 2.37·71-s + 0.692·75-s − 1.22·81-s + 0.423·89-s + 0.829·93-s + 0.406·97-s − 0.201·99-s + 0.197·103-s − 2.27·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072300840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072300840\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.73311294802687765057186922169, −7.54561336294034412073279786661, −7.17886861725990853355748964788, −6.72272822496619654178144260310, −6.15495674370883101320746381184, −5.58424399557993157488343507385, −5.14965361578058871566767036844, −4.71870190776941205025891445163, −4.05343010620277076619793237026, −3.70736921355963236223034374457, −3.31826141933768684054421304756, −2.62292055563398594746175853171, −2.40190179729436095411148612369, −1.53484161481100560074403692993, −0.55738400070268221626387585017,
0.55738400070268221626387585017, 1.53484161481100560074403692993, 2.40190179729436095411148612369, 2.62292055563398594746175853171, 3.31826141933768684054421304756, 3.70736921355963236223034374457, 4.05343010620277076619793237026, 4.71870190776941205025891445163, 5.14965361578058871566767036844, 5.58424399557993157488343507385, 6.15495674370883101320746381184, 6.72272822496619654178144260310, 7.17886861725990853355748964788, 7.54561336294034412073279786661, 7.73311294802687765057186922169
