| L(s) = 1 | + 3-s + 9-s − 6·11-s − 2·13-s − 4·23-s + 25-s + 27-s − 6·33-s − 2·37-s − 2·39-s − 8·47-s − 6·49-s − 2·59-s + 4·61-s − 4·69-s − 16·71-s − 8·73-s + 75-s + 81-s − 12·83-s + 4·97-s − 6·99-s + 8·107-s + 4·109-s − 2·111-s − 2·117-s + 14·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.04·33-s − 0.328·37-s − 0.320·39-s − 1.16·47-s − 6/7·49-s − 0.260·59-s + 0.512·61-s − 0.481·69-s − 1.89·71-s − 0.936·73-s + 0.115·75-s + 1/9·81-s − 1.31·83-s + 0.406·97-s − 0.603·99-s + 0.773·107-s + 0.383·109-s − 0.189·111-s − 0.184·117-s + 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.742712558744002350260619090470, −8.536103907543821129454387229852, −7.929281054613052383393998170601, −7.61821270029823283486198862923, −7.19104594571292042836578137181, −6.56863084021586537982840850993, −5.90148277844004930868704873342, −5.44161428144835670905246681027, −4.79774534788073660052653117779, −4.45584175053390178669613747232, −3.55386146285483021693953442820, −2.96065759868694267119537321851, −2.43788296445631663518446955536, −1.65301994477207494420567951959, 0,
1.65301994477207494420567951959, 2.43788296445631663518446955536, 2.96065759868694267119537321851, 3.55386146285483021693953442820, 4.45584175053390178669613747232, 4.79774534788073660052653117779, 5.44161428144835670905246681027, 5.90148277844004930868704873342, 6.56863084021586537982840850993, 7.19104594571292042836578137181, 7.61821270029823283486198862923, 7.929281054613052383393998170601, 8.536103907543821129454387229852, 8.742712558744002350260619090470
