| L(s) = 1 | + 2-s − 5-s − 5·7-s − 8-s − 10-s + 11-s + 2·13-s − 5·14-s − 16-s + 2·17-s − 2·19-s + 22-s − 2·23-s + 5·25-s + 2·26-s + 14·29-s + 3·31-s + 2·34-s + 5·35-s + 2·37-s − 2·38-s + 40-s + 16·41-s − 16·43-s − 2·46-s + 18·49-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.458·19-s + 0.213·22-s − 0.417·23-s + 25-s + 0.392·26-s + 2.59·29-s + 0.538·31-s + 0.342·34-s + 0.845·35-s + 0.328·37-s − 0.324·38-s + 0.158·40-s + 2.49·41-s − 2.43·43-s − 0.294·46-s + 18/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.205984339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.205984339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.622093894491467024493408415294, −9.221952887733560576221555846520, −8.780287563412957096808628831230, −8.461407271189260207952467164340, −7.82956979160123545320230786421, −7.81227564789584110571624848432, −6.83000118405512655017500408044, −6.58968846602353952967391467424, −6.37125303134915420614941992450, −6.23744926382727969879220395502, −5.24749312868986140885880679135, −5.23622599708474796615300821603, −4.45595912953395142747361170605, −4.11049392129623721090012020660, −3.46671442364925762610648445140, −3.46351819126427261670921514174, −2.55948386342743171777712658168, −2.54380968241671502048230804923, −1.13865398572604169943977465444, −0.59194156946852427616030634387,
0.59194156946852427616030634387, 1.13865398572604169943977465444, 2.54380968241671502048230804923, 2.55948386342743171777712658168, 3.46351819126427261670921514174, 3.46671442364925762610648445140, 4.11049392129623721090012020660, 4.45595912953395142747361170605, 5.23622599708474796615300821603, 5.24749312868986140885880679135, 6.23744926382727969879220395502, 6.37125303134915420614941992450, 6.58968846602353952967391467424, 6.83000118405512655017500408044, 7.81227564789584110571624848432, 7.82956979160123545320230786421, 8.461407271189260207952467164340, 8.780287563412957096808628831230, 9.221952887733560576221555846520, 9.622093894491467024493408415294
