| L(s) = 1 | + 2-s + 5-s + 4·7-s − 8-s + 10-s − 11-s + 14·13-s + 4·14-s − 16-s − 4·17-s − 19-s − 22-s + 23-s + 14·26-s + 16·29-s − 6·31-s − 4·34-s + 4·35-s + 3·37-s − 38-s − 40-s − 18·41-s − 8·43-s + 46-s − 3·47-s + 9·49-s − 53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 3.88·13-s + 1.06·14-s − 1/4·16-s − 0.970·17-s − 0.229·19-s − 0.213·22-s + 0.208·23-s + 2.74·26-s + 2.97·29-s − 1.07·31-s − 0.685·34-s + 0.676·35-s + 0.493·37-s − 0.162·38-s − 0.158·40-s − 2.81·41-s − 1.21·43-s + 0.147·46-s − 0.437·47-s + 9/7·49-s − 0.137·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.024226186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.024226186\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−10.81969285282630764680299922057, −10.69801583489640963834149074405, −10.02248943322612552983349154037, −9.572632805190784027739284739037, −8.666585487367050046528870742845, −8.534316096540877480763042876204, −8.312231360735852369355269882641, −8.202568889945152015460403992989, −6.97359734140330362362674046285, −6.59300376078337202567710398881, −6.41903359710852254102492728028, −5.69958069259223045967488953392, −5.31175734659128630474313456349, −4.85809735754376959928951563893, −4.26974582223375506078480735217, −3.77626609043496592640883211206, −3.33724876461643906643390240718, −2.47701075094169862455421068176, −1.56088455202916376653480763358, −1.21355741267945689276250842244,
1.21355741267945689276250842244, 1.56088455202916376653480763358, 2.47701075094169862455421068176, 3.33724876461643906643390240718, 3.77626609043496592640883211206, 4.26974582223375506078480735217, 4.85809735754376959928951563893, 5.31175734659128630474313456349, 5.69958069259223045967488953392, 6.41903359710852254102492728028, 6.59300376078337202567710398881, 6.97359734140330362362674046285, 8.202568889945152015460403992989, 8.312231360735852369355269882641, 8.534316096540877480763042876204, 8.666585487367050046528870742845, 9.572632805190784027739284739037, 10.02248943322612552983349154037, 10.69801583489640963834149074405, 10.81969285282630764680299922057
