| L(s) = 1 | − 3-s − 2·4-s + 9-s + 2·12-s + 4·16-s − 8·19-s − 25-s − 27-s + 8·31-s − 2·36-s − 4·48-s + 11·49-s + 8·57-s − 14·61-s − 8·64-s − 16·67-s + 10·73-s + 75-s + 16·76-s − 16·79-s + 81-s − 8·93-s + 2·100-s − 16·103-s + 2·108-s + 5·121-s − 16·124-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s + 16-s − 1.83·19-s − 1/5·25-s − 0.192·27-s + 1.43·31-s − 1/3·36-s − 0.577·48-s + 11/7·49-s + 1.05·57-s − 1.79·61-s − 64-s − 1.95·67-s + 1.17·73-s + 0.115·75-s + 1.83·76-s − 1.80·79-s + 1/9·81-s − 0.829·93-s + 1/5·100-s − 1.57·103-s + 0.192·108-s + 5/11·121-s − 1.43·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.904576061097360707651138215797, −8.617901304990678264998953970757, −8.188857398276588140156519015573, −7.57719463711681801958954569696, −7.09629991703813270871692542563, −6.36183849961441221182971644441, −6.03921249310516027590630964803, −5.55085907842215091742630411051, −4.76799882206476029917443649366, −4.43575170842274105456443254671, −4.01381235853638191869941790465, −3.16673931725937276916292992997, −2.32579572587618680024091876188, −1.25013518041851019608853334306, 0,
1.25013518041851019608853334306, 2.32579572587618680024091876188, 3.16673931725937276916292992997, 4.01381235853638191869941790465, 4.43575170842274105456443254671, 4.76799882206476029917443649366, 5.55085907842215091742630411051, 6.03921249310516027590630964803, 6.36183849961441221182971644441, 7.09629991703813270871692542563, 7.57719463711681801958954569696, 8.188857398276588140156519015573, 8.617901304990678264998953970757, 8.904576061097360707651138215797
