| L(s) = 1 | − 22·19-s − 26·31-s − 71·49-s − 242·61-s − 22·79-s − 286·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 1.15·19-s − 0.838·31-s − 1.44·49-s − 3.96·61-s − 0.278·79-s − 2.62·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-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 − 0.863·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2563150977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2563150977\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 71 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2591 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 121 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5906 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9791 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9071 T^{2} + p^{4} T^{4} \) |
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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
−9.073557887026736296318514264351, −8.395977746356363626235915588024, −7.980895720707235161431478747590, −7.932706912650293091375922898602, −7.31456573630673369569300113278, −6.78965143778853151482423737410, −6.77217983209893791281833019925, −6.01292603445138993609531518793, −5.89717649137041413326656936094, −5.47603210974417586962935332055, −4.67347215867287830189768132569, −4.65592953884546285833663825212, −4.21993829684377790170927863714, −3.48939822360084130558549362234, −3.27802099989726504563125324409, −2.70998795229766346619217734212, −2.09066974051894200122882418212, −1.68388232994528866587379792820, −1.11915180805950267476258186426, −0.12051863085851703018968819163,
0.12051863085851703018968819163, 1.11915180805950267476258186426, 1.68388232994528866587379792820, 2.09066974051894200122882418212, 2.70998795229766346619217734212, 3.27802099989726504563125324409, 3.48939822360084130558549362234, 4.21993829684377790170927863714, 4.65592953884546285833663825212, 4.67347215867287830189768132569, 5.47603210974417586962935332055, 5.89717649137041413326656936094, 6.01292603445138993609531518793, 6.77217983209893791281833019925, 6.78965143778853151482423737410, 7.31456573630673369569300113278, 7.932706912650293091375922898602, 7.980895720707235161431478747590, 8.395977746356363626235915588024, 9.073557887026736296318514264351
