| L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 3·9-s − 5·11-s − 2·13-s + 4·15-s + 5·17-s − 4·19-s + 2·21-s + 7·23-s + 3·25-s − 4·27-s + 6·29-s − 6·31-s + 10·33-s + 2·35-s − 7·37-s + 4·39-s + 15·41-s + 8·43-s − 6·45-s + 10·47-s − 3·49-s − 10·51-s + 15·53-s + 10·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 9-s − 1.50·11-s − 0.554·13-s + 1.03·15-s + 1.21·17-s − 0.917·19-s + 0.436·21-s + 1.45·23-s + 3/5·25-s − 0.769·27-s + 1.11·29-s − 1.07·31-s + 1.74·33-s + 0.338·35-s − 1.15·37-s + 0.640·39-s + 2.34·41-s + 1.21·43-s − 0.894·45-s + 1.45·47-s − 3/7·49-s − 1.40·51-s + 2.06·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9707426381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9707426381\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + T - 114 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 150 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 17 T + 174 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.663767955358178597932766920478, −9.307358576304214997750921444045, −8.666376700763651782389743663831, −8.625990134542515005679825411860, −7.74419814236012157091093863671, −7.60486205789568816462047807885, −7.18255982108199440756775595537, −7.09378670714638701836806674232, −6.20554913163162826846281664299, −5.98759851001109000458512439279, −5.53263249763125471571708836931, −5.10559675455828148694822898327, −4.63630331963000520860375565331, −4.43060918439962694466860238993, −3.56362766920041246015856934915, −3.36109828288864873115914066976, −2.53121763513382319380724532853, −2.19599287041094763525578487798, −0.894035209469996076943091148765, −0.56769292784622570486080122434,
0.56769292784622570486080122434, 0.894035209469996076943091148765, 2.19599287041094763525578487798, 2.53121763513382319380724532853, 3.36109828288864873115914066976, 3.56362766920041246015856934915, 4.43060918439962694466860238993, 4.63630331963000520860375565331, 5.10559675455828148694822898327, 5.53263249763125471571708836931, 5.98759851001109000458512439279, 6.20554913163162826846281664299, 7.09378670714638701836806674232, 7.18255982108199440756775595537, 7.60486205789568816462047807885, 7.74419814236012157091093863671, 8.625990134542515005679825411860, 8.666376700763651782389743663831, 9.307358576304214997750921444045, 9.663767955358178597932766920478
