| L(s) = 1 | − 4-s + 6·9-s − 4·11-s + 16-s + 16·19-s − 2·29-s − 2·31-s − 6·36-s + 4·41-s + 4·44-s + 14·49-s + 30·59-s + 22·61-s − 64-s − 12·71-s − 16·76-s − 16·79-s + 27·81-s + 32·89-s − 24·99-s − 18·101-s + 12·109-s + 2·116-s − 10·121-s + 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s − 1.20·11-s + 1/4·16-s + 3.67·19-s − 0.371·29-s − 0.359·31-s − 36-s + 0.624·41-s + 0.603·44-s + 2·49-s + 3.90·59-s + 2.81·61-s − 1/8·64-s − 1.42·71-s − 1.83·76-s − 1.80·79-s + 3·81-s + 3.39·89-s − 2.41·99-s − 1.79·101-s + 1.14·109-s + 0.185·116-s − 0.909·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.889804936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.889804936\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.699428102399344391153195923319, −9.585857017823545708884463154615, −8.874557882551005687964720101931, −8.683370479254036510011848552499, −7.83898261222196686134467131032, −7.74031506404793690908415237653, −7.23158941478599447701630298295, −7.17793372841849974101854806757, −6.67098056811928879251131777832, −5.78514499074730611808600407151, −5.41012565428142092859575305699, −5.24444080454418888921225294927, −4.82185764215319171439433236097, −4.03191319487311635399987066311, −3.86255314370245977803417619261, −3.32860084557082410664160769087, −2.61535697780192640861583366893, −2.10667647848615049659208939170, −1.06542709336287273711984559794, −0.912206623876116998074364764163,
0.912206623876116998074364764163, 1.06542709336287273711984559794, 2.10667647848615049659208939170, 2.61535697780192640861583366893, 3.32860084557082410664160769087, 3.86255314370245977803417619261, 4.03191319487311635399987066311, 4.82185764215319171439433236097, 5.24444080454418888921225294927, 5.41012565428142092859575305699, 5.78514499074730611808600407151, 6.67098056811928879251131777832, 7.17793372841849974101854806757, 7.23158941478599447701630298295, 7.74031506404793690908415237653, 7.83898261222196686134467131032, 8.683370479254036510011848552499, 8.874557882551005687964720101931, 9.585857017823545708884463154615, 9.699428102399344391153195923319
