| L(s) = 1 | + 27·3-s − 125·5-s − 1.02e3·7-s + 729·9-s − 3.09e3·11-s − 1.30e4·13-s − 3.37e3·15-s + 1.87e3·17-s + 3.11e4·19-s − 2.77e4·21-s + 3.32e4·23-s + 1.56e4·25-s + 1.96e4·27-s − 2.13e5·29-s + 1.72e5·31-s − 8.35e4·33-s + 1.28e5·35-s + 2.74e4·37-s − 3.51e5·39-s + 5.32e5·41-s + 9.11e5·43-s − 9.11e4·45-s + 7.32e5·47-s + 2.33e5·49-s + 5.07e4·51-s + 4.09e5·53-s + 3.87e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.701·11-s − 1.64·13-s − 0.258·15-s + 0.0927·17-s + 1.04·19-s − 0.654·21-s + 0.570·23-s + 1/5·25-s + 0.192·27-s − 1.62·29-s + 1.04·31-s − 0.404·33-s + 0.506·35-s + 0.0890·37-s − 0.949·39-s + 1.20·41-s + 1.74·43-s − 0.149·45-s + 1.02·47-s + 0.283·49-s + 0.0535·51-s + 0.377·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.476384986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476384986\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 + p^{3} T \) |
| good | 7 | \( 1 + 1028 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3096 T + p^{7} T^{2} \) |
| 13 | \( 1 + 13030 T + p^{7} T^{2} \) |
| 17 | \( 1 - 1878 T + p^{7} T^{2} \) |
| 19 | \( 1 - 31180 T + p^{7} T^{2} \) |
| 23 | \( 1 - 33288 T + p^{7} T^{2} \) |
| 29 | \( 1 + 213054 T + p^{7} T^{2} \) |
| 31 | \( 1 - 172696 T + p^{7} T^{2} \) |
| 37 | \( 1 - 27434 T + p^{7} T^{2} \) |
| 41 | \( 1 - 532650 T + p^{7} T^{2} \) |
| 43 | \( 1 - 911908 T + p^{7} T^{2} \) |
| 47 | \( 1 - 732648 T + p^{7} T^{2} \) |
| 53 | \( 1 - 409074 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1508136 T + p^{7} T^{2} \) |
| 61 | \( 1 + 302578 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1254332 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4781280 T + p^{7} T^{2} \) |
| 73 | \( 1 + 502414 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1991368 T + p^{7} T^{2} \) |
| 83 | \( 1 - 8099268 T + p^{7} T^{2} \) |
| 89 | \( 1 - 7487970 T + p^{7} T^{2} \) |
| 97 | \( 1 + 17172574 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.73069141920775563833848045355, −9.712303822604943575191740005975, −9.160671784348119497256149178272, −7.65133061856528015950639969011, −7.26327046809883843238270533911, −5.77808133587093410218856181730, −4.53608779612770015106014891366, −3.23781187010243431790095258037, −2.45812486939073937552219476807, −0.57868870811026314625490907777,
0.57868870811026314625490907777, 2.45812486939073937552219476807, 3.23781187010243431790095258037, 4.53608779612770015106014891366, 5.77808133587093410218856181730, 7.26327046809883843238270533911, 7.65133061856528015950639969011, 9.160671784348119497256149178272, 9.712303822604943575191740005975, 10.73069141920775563833848045355
