L(s) = 1 | − 3·3-s − 2·4-s − 4·7-s + 6·9-s + 6·12-s + 6·19-s + 12·21-s − 9·27-s + 8·28-s − 15·31-s − 12·36-s − 11·37-s + 10·43-s + 9·49-s − 18·57-s + 27·61-s − 24·63-s + 8·64-s − 16·67-s + 3·73-s − 12·76-s − 17·79-s + 9·81-s − 24·84-s + 45·93-s − 27·103-s + 18·108-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 4-s − 1.51·7-s + 2·9-s + 1.73·12-s + 1.37·19-s + 2.61·21-s − 1.73·27-s + 1.51·28-s − 2.69·31-s − 2·36-s − 1.80·37-s + 1.52·43-s + 9/7·49-s − 2.38·57-s + 3.45·61-s − 3.02·63-s + 64-s − 1.95·67-s + 0.351·73-s − 1.37·76-s − 1.91·79-s + 81-s − 2.61·84-s + 4.66·93-s − 2.66·103-s + 1.73·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2191689427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2191689427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.98388937734398707745362662684, −10.70502872866695506007782725938, −10.16439879762252641259284961504, −9.735795588185913057371186093046, −9.549037082288771884611042367968, −8.875271833987165040728417833889, −8.773706315617616460455465209010, −7.75869065758754155126310766525, −7.11471536086813389371964565444, −7.08010138820884867212818598528, −6.49284029989436373858784559154, −5.77625279446492408405608457716, −5.48896659900045136605971722695, −5.24466650318102254395579568711, −4.51713878205467890193379286635, −3.72688113226704224038148664973, −3.69431087386852630741316490631, −2.60421819551448974455189700880, −1.39515444855475773652534463476, −0.33065262391938944937574170585,
0.33065262391938944937574170585, 1.39515444855475773652534463476, 2.60421819551448974455189700880, 3.69431087386852630741316490631, 3.72688113226704224038148664973, 4.51713878205467890193379286635, 5.24466650318102254395579568711, 5.48896659900045136605971722695, 5.77625279446492408405608457716, 6.49284029989436373858784559154, 7.08010138820884867212818598528, 7.11471536086813389371964565444, 7.75869065758754155126310766525, 8.773706315617616460455465209010, 8.875271833987165040728417833889, 9.549037082288771884611042367968, 9.735795588185913057371186093046, 10.16439879762252641259284961504, 10.70502872866695506007782725938, 10.98388937734398707745362662684