| L(s) = 1 | + 3-s + 2·5-s − 2·11-s + 8·13-s + 2·15-s + 6·17-s + 8·19-s + 6·23-s + 5·25-s − 27-s − 20·29-s + 4·31-s − 2·33-s − 6·37-s + 8·39-s + 12·41-s + 8·43-s + 8·47-s + 6·51-s − 2·53-s − 4·55-s + 8·57-s − 4·59-s − 8·61-s + 16·65-s + 8·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.603·11-s + 2.21·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s + 1.25·23-s + 25-s − 0.192·27-s − 3.71·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 1.28·39-s + 1.87·41-s + 1.21·43-s + 1.16·47-s + 0.840·51-s − 0.274·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s − 1.02·61-s + 1.98·65-s + 0.977·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.278665351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.278665351\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−10.73965459330473160920261018699, −10.72778583873254232817321505124, −9.731319955720249993534157848566, −9.688946276586426160804765139668, −9.142962783455452026301481432854, −8.835877260878964498105470781262, −8.412426757454182326040029176675, −7.63984826694279960957230548952, −7.45629129092727535273265018336, −7.14265581430412680312797508121, −6.11892232742314159672942746422, −5.85860392646274529244502887464, −5.47156165595483659599096975733, −5.18891765822095017483420416119, −4.01716907375447724638111147930, −3.78106439838390656310252233410, −2.91728336039430706781360995339, −2.82157880809136794235737109534, −1.48454647706044124862202797041, −1.23553727931513015429673319306,
1.23553727931513015429673319306, 1.48454647706044124862202797041, 2.82157880809136794235737109534, 2.91728336039430706781360995339, 3.78106439838390656310252233410, 4.01716907375447724638111147930, 5.18891765822095017483420416119, 5.47156165595483659599096975733, 5.85860392646274529244502887464, 6.11892232742314159672942746422, 7.14265581430412680312797508121, 7.45629129092727535273265018336, 7.63984826694279960957230548952, 8.412426757454182326040029176675, 8.835877260878964498105470781262, 9.142962783455452026301481432854, 9.688946276586426160804765139668, 9.731319955720249993534157848566, 10.72778583873254232817321505124, 10.73965459330473160920261018699
