| L(s) = 1 | + 4·7-s − 8·17-s + 16·23-s + 6·25-s + 12·31-s + 24·41-s − 16·47-s − 2·49-s − 4·73-s + 28·79-s − 16·89-s − 4·97-s + 28·103-s − 16·113-s − 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.94·17-s + 3.33·23-s + 6/5·25-s + 2.15·31-s + 3.74·41-s − 2.33·47-s − 2/7·49-s − 0.468·73-s + 3.15·79-s − 1.69·89-s − 0.406·97-s + 2.75·103-s − 1.50·113-s − 2.93·119-s + 6/11·121-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 + 0.0798·157-s + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.818959941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.818959941\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.071770219641167348781814342603, −8.911966787337030074926601960844, −8.291075298961237783859183623115, −8.218781264440375284238641931958, −7.70541239741409786635088172892, −7.28633607395926117884851400993, −6.81602000442399146212800856492, −6.52753032453984109426690266289, −6.29062222874349048401808275412, −5.50937409345539298850416806282, −5.07877775972606713572339068965, −4.75462772857192464273749896097, −4.47799024983447709974739020709, −4.28937179762605532913910805486, −3.21736699297368596350936873975, −2.98823899997234689004799977554, −2.44386756708720472514946110427, −1.92630733709849888654509153709, −1.10692434359277039338905228255, −0.814333099439137324094436879149,
0.814333099439137324094436879149, 1.10692434359277039338905228255, 1.92630733709849888654509153709, 2.44386756708720472514946110427, 2.98823899997234689004799977554, 3.21736699297368596350936873975, 4.28937179762605532913910805486, 4.47799024983447709974739020709, 4.75462772857192464273749896097, 5.07877775972606713572339068965, 5.50937409345539298850416806282, 6.29062222874349048401808275412, 6.52753032453984109426690266289, 6.81602000442399146212800856492, 7.28633607395926117884851400993, 7.70541239741409786635088172892, 8.218781264440375284238641931958, 8.291075298961237783859183623115, 8.911966787337030074926601960844, 9.071770219641167348781814342603
