| L(s) = 1 | − 4-s − 7-s − 3·9-s − 6·13-s + 16-s + 8·25-s + 28-s − 6·31-s + 3·36-s + 8·37-s + 8·43-s + 49-s + 6·52-s − 12·61-s + 3·63-s − 64-s + 4·67-s + 17·73-s − 2·79-s + 9·81-s + 6·91-s − 8·100-s − 18·103-s − 28·109-s − 112-s + 18·117-s + 4·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.377·7-s − 9-s − 1.66·13-s + 1/4·16-s + 8/5·25-s + 0.188·28-s − 1.07·31-s + 1/2·36-s + 1.31·37-s + 1.21·43-s + 1/7·49-s + 0.832·52-s − 1.53·61-s + 0.377·63-s − 1/8·64-s + 0.488·67-s + 1.98·73-s − 0.225·79-s + 81-s + 0.628·91-s − 4/5·100-s − 1.77·103-s − 2.68·109-s − 0.0944·112-s + 1.66·117-s + 4/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 16 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.054164276153657025216058182678, −7.49295013671548862296143041230, −7.13641414814088792697999514613, −6.66959596408741876407724833667, −6.12723577952151804587764415774, −5.66161128027398707240359691807, −5.17871363454874673458612847932, −4.85937608862981009111570410509, −4.28519418594894667147101211517, −3.74947567953645638022669054009, −2.96297744721935200465180770406, −2.74154791529486118242986057438, −2.07424052785995222127260664669, −0.920448659832283933640261317476, 0,
0.920448659832283933640261317476, 2.07424052785995222127260664669, 2.74154791529486118242986057438, 2.96297744721935200465180770406, 3.74947567953645638022669054009, 4.28519418594894667147101211517, 4.85937608862981009111570410509, 5.17871363454874673458612847932, 5.66161128027398707240359691807, 6.12723577952151804587764415774, 6.66959596408741876407724833667, 7.13641414814088792697999514613, 7.49295013671548862296143041230, 8.054164276153657025216058182678
