| L(s) = 1 | − 2·5-s + 2·9-s − 2·13-s − 14·17-s + 3·25-s − 12·29-s − 2·37-s + 12·41-s − 4·45-s − 2·49-s − 4·61-s + 4·65-s − 10·73-s − 5·81-s + 28·85-s + 30·89-s − 8·97-s + 4·101-s − 12·109-s − 10·113-s − 4·117-s + 4·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s − 0.554·13-s − 3.39·17-s + 3/5·25-s − 2.22·29-s − 0.328·37-s + 1.87·41-s − 0.596·45-s − 2/7·49-s − 0.512·61-s + 0.496·65-s − 1.17·73-s − 5/9·81-s + 3.03·85-s + 3.17·89-s − 0.812·97-s + 0.398·101-s − 1.14·109-s − 0.940·113-s − 0.369·117-s + 4/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6340793570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6340793570\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.889663408994127642205088075233, −7.61235167985819900020846324929, −7.28742789208806835868762617163, −6.78125754641445802626921354443, −6.48951780063079708387665227484, −5.95457467179700615796875041208, −5.30720441303741072337697813419, −4.74986597775864058336344224182, −4.34264987642983189559042284408, −4.09053621351507887332446601406, −3.55860075792206233293256152311, −2.70426791533871740541996668827, −2.22425805433938852271003824286, −1.64757392421881940461930784156, −0.34823057566950090126329869440,
0.34823057566950090126329869440, 1.64757392421881940461930784156, 2.22425805433938852271003824286, 2.70426791533871740541996668827, 3.55860075792206233293256152311, 4.09053621351507887332446601406, 4.34264987642983189559042284408, 4.74986597775864058336344224182, 5.30720441303741072337697813419, 5.95457467179700615796875041208, 6.48951780063079708387665227484, 6.78125754641445802626921354443, 7.28742789208806835868762617163, 7.61235167985819900020846324929, 7.889663408994127642205088075233
