| L(s) = 1 | − 2·3-s − 4-s + 9-s + 6·11-s + 2·12-s + 16-s + 6·23-s + 4·27-s − 4·31-s − 12·33-s − 36-s − 6·37-s − 6·44-s − 2·48-s + 10·49-s + 18·53-s − 6·59-s − 64-s − 12·67-s − 12·69-s − 12·71-s − 11·81-s − 30·89-s − 6·92-s + 8·93-s + 24·97-s + 6·99-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1/3·9-s + 1.80·11-s + 0.577·12-s + 1/4·16-s + 1.25·23-s + 0.769·27-s − 0.718·31-s − 2.08·33-s − 1/6·36-s − 0.986·37-s − 0.904·44-s − 0.288·48-s + 10/7·49-s + 2.47·53-s − 0.781·59-s − 1/8·64-s − 1.46·67-s − 1.44·69-s − 1.42·71-s − 1.22·81-s − 3.17·89-s − 0.625·92-s + 0.829·93-s + 2.43·97-s + 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
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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
−7.24531710256254297085778010128, −6.99914246522674329155880281085, −6.53560756898451972211407733715, −6.02112806025489502871655396251, −5.73570796992316738046549928902, −5.39361112397672692398057116978, −4.77674336246898332049753361281, −4.48411979613718385907499329820, −3.96663400044815352195045727688, −3.53850792319081370664887559744, −2.98535621765787864816853608355, −2.22927913086969865115441291804, −1.33795924852386860532424039718, −1.00952783516717154664744262108, 0,
1.00952783516717154664744262108, 1.33795924852386860532424039718, 2.22927913086969865115441291804, 2.98535621765787864816853608355, 3.53850792319081370664887559744, 3.96663400044815352195045727688, 4.48411979613718385907499329820, 4.77674336246898332049753361281, 5.39361112397672692398057116978, 5.73570796992316738046549928902, 6.02112806025489502871655396251, 6.53560756898451972211407733715, 6.99914246522674329155880281085, 7.24531710256254297085778010128
