| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s − 4·11-s − 5·13-s + 4·15-s − 3·17-s + 2·19-s + 2·23-s − 7·25-s − 10·27-s + 5·29-s − 4·31-s + 8·33-s + 5·37-s + 10·39-s − 3·41-s + 4·43-s − 6·45-s − 12·47-s + 7·49-s + 6·51-s − 26·53-s + 8·55-s − 4·57-s − 12·59-s − 7·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s − 1.20·11-s − 1.38·13-s + 1.03·15-s − 0.727·17-s + 0.458·19-s + 0.417·23-s − 7/5·25-s − 1.92·27-s + 0.928·29-s − 0.718·31-s + 1.39·33-s + 0.821·37-s + 1.60·39-s − 0.468·41-s + 0.609·43-s − 0.894·45-s − 1.75·47-s + 49-s + 0.840·51-s − 3.57·53-s + 1.07·55-s − 0.529·57-s − 1.56·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.755939481145026765633752700028, −9.630209143603333530824364613625, −9.629696918347909247258172870597, −8.660443607351934605163449136980, −8.034811696720701676218661773854, −7.77262418328361040576312245365, −7.54637921581084797412542331439, −7.07370601698036457028517533921, −6.33611272842059421796395011230, −6.24332479928020471850298565135, −5.40004752762507478010660547365, −5.11622260764358190081503364725, −4.74305986137614045969954477516, −4.21262761918945774011885828446, −3.65562182532463787298423412756, −2.93186251521172723495321887796, −2.30922507337412025737703628465, −1.50748275186166494399882492118, 0, 0,
1.50748275186166494399882492118, 2.30922507337412025737703628465, 2.93186251521172723495321887796, 3.65562182532463787298423412756, 4.21262761918945774011885828446, 4.74305986137614045969954477516, 5.11622260764358190081503364725, 5.40004752762507478010660547365, 6.24332479928020471850298565135, 6.33611272842059421796395011230, 7.07370601698036457028517533921, 7.54637921581084797412542331439, 7.77262418328361040576312245365, 8.034811696720701676218661773854, 8.660443607351934605163449136980, 9.629696918347909247258172870597, 9.630209143603333530824364613625, 9.755939481145026765633752700028
