| L(s) = 1 | + 6·7-s + 14·13-s + 38·19-s + 42·25-s + 20·31-s + 126·37-s + 100·43-s − 71·49-s + 158·61-s − 154·67-s − 34·73-s + 22·79-s + 84·91-s − 194·97-s − 266·103-s + 260·109-s − 150·121-s + 127-s + 131-s + 228·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 6/7·7-s + 1.07·13-s + 2·19-s + 1.67·25-s + 0.645·31-s + 3.40·37-s + 2.32·43-s − 1.44·49-s + 2.59·61-s − 2.29·67-s − 0.465·73-s + 0.278·79-s + 0.923·91-s − 2·97-s − 2.58·103-s + 2.38·109-s − 1.23·121-s + 0.00787·127-s + 0.00763·131-s + 12/7·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.561354123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.561354123\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 42 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 150 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 378 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1050 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 910 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 63 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2618 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2838 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 79 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 77 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3810 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12210 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14042 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
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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
−11.07366907213627216276494884233, −11.01172810815210679482574833038, −10.27651351664775301952193232692, −9.722822120593600398110635603888, −9.420185475895686059478185989022, −8.911317567894948410404151425860, −8.340198672416212487777829865488, −8.044900324965701839433663689183, −7.42140747333309007734333301289, −7.22256247415856237216283521667, −6.21034666960071797937389138166, −6.15985326383474259126589112335, −5.34902038591234059966719431386, −4.93637416839061581395202334458, −4.32239105791055978389725079863, −3.81443249440583985393893945340, −2.88903901119787943544455682428, −2.59849526136789794958576176672, −1.15920659640199745770837966573, −1.08234220576438506914487050996,
1.08234220576438506914487050996, 1.15920659640199745770837966573, 2.59849526136789794958576176672, 2.88903901119787943544455682428, 3.81443249440583985393893945340, 4.32239105791055978389725079863, 4.93637416839061581395202334458, 5.34902038591234059966719431386, 6.15985326383474259126589112335, 6.21034666960071797937389138166, 7.22256247415856237216283521667, 7.42140747333309007734333301289, 8.044900324965701839433663689183, 8.340198672416212487777829865488, 8.911317567894948410404151425860, 9.420185475895686059478185989022, 9.722822120593600398110635603888, 10.27651351664775301952193232692, 11.01172810815210679482574833038, 11.07366907213627216276494884233
