L(s) = 1 | − 2·4-s − 16·13-s + 4·16-s + 32·19-s + 32·25-s − 88·31-s − 68·37-s − 80·43-s + 32·52-s − 100·61-s − 8·64-s + 16·67-s + 32·73-s − 64·76-s − 152·79-s − 352·97-s − 64·100-s + 56·103-s − 46·121-s + 176·124-s + 127-s + 131-s + 137-s + 139-s + 136·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.23·13-s + 1/4·16-s + 1.68·19-s + 1.27·25-s − 2.83·31-s − 1.83·37-s − 1.86·43-s + 8/13·52-s − 1.63·61-s − 1/8·64-s + 0.238·67-s + 0.438·73-s − 0.842·76-s − 1.92·79-s − 3.62·97-s − 0.639·100-s + 0.543·103-s − 0.380·121-s + 1.41·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.918·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7537953800\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7537953800\) |
\(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1664 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4160 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15680 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 176 T + p^{2} T^{2} )^{2} \) |
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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
−10.30043516936086576043929229061, −9.615085954447740487433903839774, −9.406925528344981426811008520087, −8.784479815182659277731965580058, −8.733354996297716235976672692723, −7.999945832806824503288140739230, −7.50619857545755213365572744010, −7.19682686579861573137408694641, −6.92627384444924372589096723994, −6.30597137565107976123829190076, −5.49657973238433766426055376084, −5.21975034065573553757766612376, −5.10652681356493178402301024112, −4.34906018890944444377569569981, −3.77125762750569263611071986298, −3.15559971172624801595723374441, −2.91145975929704736032018073339, −1.85059320603374405960128428960, −1.42577528943012396507964750190, −0.28912113765826791572762518793,
0.28912113765826791572762518793, 1.42577528943012396507964750190, 1.85059320603374405960128428960, 2.91145975929704736032018073339, 3.15559971172624801595723374441, 3.77125762750569263611071986298, 4.34906018890944444377569569981, 5.10652681356493178402301024112, 5.21975034065573553757766612376, 5.49657973238433766426055376084, 6.30597137565107976123829190076, 6.92627384444924372589096723994, 7.19682686579861573137408694641, 7.50619857545755213365572744010, 7.999945832806824503288140739230, 8.733354996297716235976672692723, 8.784479815182659277731965580058, 9.406925528344981426811008520087, 9.615085954447740487433903839774, 10.30043516936086576043929229061