| L(s) = 1 | − 4-s + 2·9-s − 12·11-s + 16-s − 4·19-s + 2·29-s − 14·31-s − 2·36-s − 24·41-s + 12·44-s + 10·49-s + 18·59-s − 2·61-s − 64-s − 24·71-s + 4·76-s − 16·79-s − 5·81-s + 12·89-s − 24·99-s + 6·101-s + 8·109-s − 2·116-s + 86·121-s + 14·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s − 3.61·11-s + 1/4·16-s − 0.917·19-s + 0.371·29-s − 2.51·31-s − 1/3·36-s − 3.74·41-s + 1.80·44-s + 10/7·49-s + 2.34·59-s − 0.256·61-s − 1/8·64-s − 2.84·71-s + 0.458·76-s − 1.80·79-s − 5/9·81-s + 1.27·89-s − 2.41·99-s + 0.597·101-s + 0.766·109-s − 0.185·116-s + 7.81·121-s + 1.25·124-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−9.227711834250440085016337198682, −8.748367034588930178138098096136, −8.601804173317114481841567989198, −8.115711287289803492281037559368, −7.73932642641320336328375962718, −7.30899608790696544734888275504, −7.10600051731738631577252129864, −6.56296677293455952721598660339, −5.68196533244516170791628641172, −5.51672628587367986842256871790, −5.29642208402497731274704941599, −4.59916385054231776570974859258, −4.50437246840661578070239812349, −3.47234528854097161946803218723, −3.40470031884530021782360939367, −2.45875730195634571050949491854, −2.27091093971533352857860902265, −1.47852189572173631258395816757, 0, 0,
1.47852189572173631258395816757, 2.27091093971533352857860902265, 2.45875730195634571050949491854, 3.40470031884530021782360939367, 3.47234528854097161946803218723, 4.50437246840661578070239812349, 4.59916385054231776570974859258, 5.29642208402497731274704941599, 5.51672628587367986842256871790, 5.68196533244516170791628641172, 6.56296677293455952721598660339, 7.10600051731738631577252129864, 7.30899608790696544734888275504, 7.73932642641320336328375962718, 8.115711287289803492281037559368, 8.601804173317114481841567989198, 8.748367034588930178138098096136, 9.227711834250440085016337198682
