| L(s) = 1 | − 6·5-s − 6·9-s − 6·11-s − 4·16-s + 6·23-s + 17·25-s − 6·31-s − 8·37-s + 36·45-s + 14·47-s + 49-s − 18·53-s + 36·55-s + 16·59-s − 12·67-s − 16·71-s + 24·80-s + 27·81-s + 6·89-s + 14·97-s + 36·99-s − 8·103-s − 6·113-s − 36·115-s + 25·121-s − 18·125-s + 127-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 2·9-s − 1.80·11-s − 16-s + 1.25·23-s + 17/5·25-s − 1.07·31-s − 1.31·37-s + 5.36·45-s + 2.04·47-s + 1/7·49-s − 2.47·53-s + 4.85·55-s + 2.08·59-s − 1.46·67-s − 1.89·71-s + 2.68·80-s + 3·81-s + 0.635·89-s + 1.42·97-s + 3.61·99-s − 0.788·103-s − 0.564·113-s − 3.35·115-s + 2.27·121-s − 1.60·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−7.88305646625584781075065151406, −7.43563929744309328324545511818, −7.38318915968245477906353475134, −6.75747750884296153616010532091, −6.10493668511615122876893578729, −5.41120349293735009167808988128, −5.27883313770925671699270459709, −4.64299598156704998773380783331, −4.22570906854938018403998871989, −3.57094110488123461480235960376, −3.11744276397134147631028514628, −2.85617269550579922001781563132, −2.11792864837681218638414366758, −0.49790947708643134938837315000, 0,
0.49790947708643134938837315000, 2.11792864837681218638414366758, 2.85617269550579922001781563132, 3.11744276397134147631028514628, 3.57094110488123461480235960376, 4.22570906854938018403998871989, 4.64299598156704998773380783331, 5.27883313770925671699270459709, 5.41120349293735009167808988128, 6.10493668511615122876893578729, 6.75747750884296153616010532091, 7.38318915968245477906353475134, 7.43563929744309328324545511818, 7.88305646625584781075065151406
