| L(s) = 1 | − 3-s + 7-s − 2·9-s + 2·13-s − 5·19-s − 21-s − 9·25-s + 5·27-s + 13·31-s + 9·37-s − 2·39-s − 8·43-s − 6·49-s + 5·57-s − 10·61-s − 2·63-s + 5·67-s − 7·73-s + 9·75-s − 5·79-s + 81-s + 2·91-s − 13·93-s − 20·97-s − 5·103-s + 109-s − 9·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.554·13-s − 1.14·19-s − 0.218·21-s − 9/5·25-s + 0.962·27-s + 2.33·31-s + 1.47·37-s − 0.320·39-s − 1.21·43-s − 6/7·49-s + 0.662·57-s − 1.28·61-s − 0.251·63-s + 0.610·67-s − 0.819·73-s + 1.03·75-s − 0.562·79-s + 1/9·81-s + 0.209·91-s − 1.34·93-s − 2.03·97-s − 0.492·103-s + 0.0957·109-s − 0.854·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.247277482818042154141291912320, −8.105579160013656166754444943734, −7.54204460293528700852536030239, −6.76699867840699566176818344069, −6.39935893507527285041068441562, −6.06288056841579588688136936120, −5.65373091756537697952504643362, −5.04120570606109310459411765754, −4.38500845819231518228312315816, −4.23660685627201790575233059507, −3.31602836143344191565160898648, −2.75467066899842230371875721653, −2.05084442827772284983078581816, −1.18731963404767717116883968044, 0,
1.18731963404767717116883968044, 2.05084442827772284983078581816, 2.75467066899842230371875721653, 3.31602836143344191565160898648, 4.23660685627201790575233059507, 4.38500845819231518228312315816, 5.04120570606109310459411765754, 5.65373091756537697952504643362, 6.06288056841579588688136936120, 6.39935893507527285041068441562, 6.76699867840699566176818344069, 7.54204460293528700852536030239, 8.105579160013656166754444943734, 8.247277482818042154141291912320
