| L(s) = 1 | − 4·2-s + 12·4-s − 6·5-s − 32·8-s + 24·10-s + 80·16-s + 30·17-s − 72·20-s + 11·25-s + 82·29-s − 192·32-s − 120·34-s + 94·37-s + 192·40-s + 62·41-s − 44·50-s + 180·53-s − 328·58-s + 98·61-s + 448·64-s + 360·68-s + 206·73-s − 376·74-s − 480·80-s − 81·81-s − 248·82-s − 180·85-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 6/5·5-s − 4·8-s + 12/5·10-s + 5·16-s + 1.76·17-s − 3.59·20-s + 0.439·25-s + 2.82·29-s − 6·32-s − 3.52·34-s + 2.54·37-s + 24/5·40-s + 1.51·41-s − 0.879·50-s + 3.39·53-s − 5.65·58-s + 1.60·61-s + 7·64-s + 5.29·68-s + 2.82·73-s − 5.08·74-s − 6·80-s − 81-s − 3.02·82-s − 2.11·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8483449971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8483449971\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| 17 | $C_2$ | \( 1 - 30 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 - 40 T + p^{2} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 - 24 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )( 1 + 18 T + p^{2} T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 - 96 T + p^{2} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )( 1 + 130 T + p^{2} T^{2} ) \) |
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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
−11.49523215978035047100318548173, −11.00583809523909485557802262218, −10.40521579969282361548197130997, −10.22815664963447424522258945575, −9.505934135738073701996114743149, −9.465504952280010593090944704016, −8.425524515799221734912414927262, −8.360979504957234291223092086139, −7.911818463951221404782990969118, −7.57434570535738896400265962045, −6.83616152795692805220301923669, −6.71023907033619915613454968804, −5.75631362801116212626931009592, −5.49834430711532190941033127428, −4.27941471199832311719525713797, −3.74364717745532320736066235224, −2.76986156686668239553725885679, −2.56979858112938755158745772580, −1.01384940167317431649522728531, −0.795221529981571732814551486519,
0.795221529981571732814551486519, 1.01384940167317431649522728531, 2.56979858112938755158745772580, 2.76986156686668239553725885679, 3.74364717745532320736066235224, 4.27941471199832311719525713797, 5.49834430711532190941033127428, 5.75631362801116212626931009592, 6.71023907033619915613454968804, 6.83616152795692805220301923669, 7.57434570535738896400265962045, 7.911818463951221404782990969118, 8.360979504957234291223092086139, 8.425524515799221734912414927262, 9.465504952280010593090944704016, 9.505934135738073701996114743149, 10.22815664963447424522258945575, 10.40521579969282361548197130997, 11.00583809523909485557802262218, 11.49523215978035047100318548173
