| L(s) = 1 | − 2·5-s + 7-s + 2·11-s − 13-s + 12·17-s + 10·19-s − 6·23-s + 5·25-s − 8·29-s + 8·31-s − 2·35-s − 10·37-s − 8·41-s − 4·43-s + 10·47-s + 7·49-s + 8·53-s − 4·55-s − 14·59-s − 3·61-s + 2·65-s − 13·67-s − 8·71-s + 18·73-s + 2·77-s + 11·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.603·11-s − 0.277·13-s + 2.91·17-s + 2.29·19-s − 1.25·23-s + 25-s − 1.48·29-s + 1.43·31-s − 0.338·35-s − 1.64·37-s − 1.24·41-s − 0.609·43-s + 1.45·47-s + 49-s + 1.09·53-s − 0.539·55-s − 1.82·59-s − 0.384·61-s + 0.248·65-s − 1.58·67-s − 0.949·71-s + 2.10·73-s + 0.227·77-s + 1.23·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.517830772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517830772\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.305162150103503589904365861545, −8.651012364970781627165330352333, −8.048628667283817133572531591472, −7.943505167343519544669533307697, −7.61116125768748048122857742517, −7.30707336722009469309896945365, −6.97978432223247837719173769582, −6.36833677694888569057432515143, −5.91818315949575802054117332782, −5.34276472833932407969641190144, −5.32980517024360692403866744061, −4.88468629897206806418541197235, −4.16450389275578763840297775193, −3.74100327052668068585441883213, −3.44522338230579655522703000602, −3.12937001845404679352790588891, −2.51818703227741552165695300142, −1.54959302163224183759014172545, −1.30170355327052187319400668430, −0.58634323116970206231022730702,
0.58634323116970206231022730702, 1.30170355327052187319400668430, 1.54959302163224183759014172545, 2.51818703227741552165695300142, 3.12937001845404679352790588891, 3.44522338230579655522703000602, 3.74100327052668068585441883213, 4.16450389275578763840297775193, 4.88468629897206806418541197235, 5.32980517024360692403866744061, 5.34276472833932407969641190144, 5.91818315949575802054117332782, 6.36833677694888569057432515143, 6.97978432223247837719173769582, 7.30707336722009469309896945365, 7.61116125768748048122857742517, 7.943505167343519544669533307697, 8.048628667283817133572531591472, 8.651012364970781627165330352333, 9.305162150103503589904365861545
