| L(s) = 1 | − 9·13-s + 6·17-s + 8·19-s − 6·23-s + 5·25-s + 6·29-s − 2·31-s + 12·41-s + 9·43-s − 18·47-s + 11·49-s + 18·53-s + 12·59-s + 5·61-s + 13·67-s + 5·73-s − 79-s + 24·97-s + 2·103-s − 24·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.49·13-s + 1.45·17-s + 1.83·19-s − 1.25·23-s + 25-s + 1.11·29-s − 0.359·31-s + 1.87·41-s + 1.37·43-s − 2.62·47-s + 11/7·49-s + 2.47·53-s + 1.56·59-s + 0.640·61-s + 1.58·67-s + 0.585·73-s − 0.112·79-s + 2.43·97-s + 0.197·103-s − 2.29·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.952820610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952820610\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 155 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18 T + 161 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−9.022628458601438721490821584852, −8.729005477227680919121108753120, −7.994157510319160775047178642610, −7.900214878427305859215813754000, −7.42638838008225255422949644226, −7.35477575038134391865709489962, −6.66024461727532770436596887295, −6.60200220926847288805960798666, −5.64017647220150830120279371407, −5.50477250889786655422677599115, −5.25941528822827524071450067073, −4.83290734180628388147737114802, −4.12431570009246290527008043919, −4.03819059256214926568062256253, −3.07467828788431848375315418053, −3.05058615776048501198131005023, −2.29616242724933958031228319641, −2.09276069764711930499708413975, −0.852963028064141971376002733129, −0.78612103404515419227129834160,
0.78612103404515419227129834160, 0.852963028064141971376002733129, 2.09276069764711930499708413975, 2.29616242724933958031228319641, 3.05058615776048501198131005023, 3.07467828788431848375315418053, 4.03819059256214926568062256253, 4.12431570009246290527008043919, 4.83290734180628388147737114802, 5.25941528822827524071450067073, 5.50477250889786655422677599115, 5.64017647220150830120279371407, 6.60200220926847288805960798666, 6.66024461727532770436596887295, 7.35477575038134391865709489962, 7.42638838008225255422949644226, 7.900214878427305859215813754000, 7.994157510319160775047178642610, 8.729005477227680919121108753120, 9.022628458601438721490821584852
