| L(s) = 1 | − 2·5-s − 2·9-s + 8·13-s + 4·17-s + 3·25-s + 18·29-s + 2·37-s + 6·41-s + 4·45-s + 8·49-s − 2·53-s + 10·61-s − 16·65-s + 6·73-s − 5·81-s − 8·85-s − 24·89-s − 2·97-s − 6·101-s + 16·109-s + 20·113-s − 16·117-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s + 2.21·13-s + 0.970·17-s + 3/5·25-s + 3.34·29-s + 0.328·37-s + 0.937·41-s + 0.596·45-s + 8/7·49-s − 0.274·53-s + 1.28·61-s − 1.98·65-s + 0.702·73-s − 5/9·81-s − 0.867·85-s − 2.54·89-s − 0.203·97-s − 0.597·101-s + 1.53·109-s + 1.88·113-s − 1.47·117-s − 0.181·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 924800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.137996157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137996157\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.296533583720364393800681331155, −7.917694428523081632246420608897, −7.24124169584863833464419402511, −6.89644724866232670869259435139, −6.26835352319713137033856094900, −6.04359467748605384361046960263, −5.55742474183361946894638493147, −4.96153209956798236813730277290, −4.33334663627908053729846833422, −4.02407651924130544495730129803, −3.36721184401922366227950854391, −3.03228721379260342195283502069, −2.41687973347641759640668133466, −1.19539831864980693995120746423, −0.842746413262578091307252864937,
0.842746413262578091307252864937, 1.19539831864980693995120746423, 2.41687973347641759640668133466, 3.03228721379260342195283502069, 3.36721184401922366227950854391, 4.02407651924130544495730129803, 4.33334663627908053729846833422, 4.96153209956798236813730277290, 5.55742474183361946894638493147, 6.04359467748605384361046960263, 6.26835352319713137033856094900, 6.89644724866232670869259435139, 7.24124169584863833464419402511, 7.917694428523081632246420608897, 8.296533583720364393800681331155
