| L(s) = 1 | + 25·5-s − 218·7-s − 480·11-s − 622·13-s − 186·17-s + 1.20e3·19-s − 3.18e3·23-s + 625·25-s − 5.52e3·29-s − 9.35e3·31-s − 5.45e3·35-s + 5.61e3·37-s + 1.43e4·41-s + 370·43-s + 1.61e4·47-s + 3.07e4·49-s + 4.37e3·53-s − 1.20e4·55-s − 1.17e4·59-s + 1.32e4·61-s − 1.55e4·65-s + 1.15e4·67-s − 2.95e4·71-s + 3.36e4·73-s + 1.04e5·77-s − 3.12e4·79-s − 3.84e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.68·7-s − 1.19·11-s − 1.02·13-s − 0.156·17-s + 0.765·19-s − 1.25·23-s + 1/5·25-s − 1.22·29-s − 1.74·31-s − 0.752·35-s + 0.674·37-s + 1.33·41-s + 0.0305·43-s + 1.06·47-s + 1.82·49-s + 0.213·53-s − 0.534·55-s − 0.439·59-s + 0.454·61-s − 0.456·65-s + 0.314·67-s − 0.695·71-s + 0.740·73-s + 2.01·77-s − 0.562·79-s − 0.612·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7257912654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7257912654\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 218 T + p^{5} T^{2} \) |
| 11 | \( 1 + 480 T + p^{5} T^{2} \) |
| 13 | \( 1 + 622 T + p^{5} T^{2} \) |
| 17 | \( 1 + 186 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1204 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3186 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9356 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5618 T + p^{5} T^{2} \) |
| 41 | \( 1 - 14394 T + p^{5} T^{2} \) |
| 43 | \( 1 - 370 T + p^{5} T^{2} \) |
| 47 | \( 1 - 16146 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4374 T + p^{5} T^{2} \) |
| 59 | \( 1 + 11748 T + p^{5} T^{2} \) |
| 61 | \( 1 - 13202 T + p^{5} T^{2} \) |
| 67 | \( 1 - 11542 T + p^{5} T^{2} \) |
| 71 | \( 1 + 29532 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33698 T + p^{5} T^{2} \) |
| 79 | \( 1 + 31208 T + p^{5} T^{2} \) |
| 83 | \( 1 + 38466 T + p^{5} T^{2} \) |
| 89 | \( 1 + 119514 T + p^{5} T^{2} \) |
| 97 | \( 1 - 94658 T + p^{5} T^{2} \) |
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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
−9.690816688234801809005562874992, −9.098566798818769925975626306144, −7.68980610564291871131544125167, −7.13290944209961654065967088343, −5.94194987005716777417466882588, −5.43562095897497227288491159121, −4.02619197900537234653869128669, −2.95440724832837064510945733095, −2.15999979740506238796366021183, −0.36848009909611478170979415846,
0.36848009909611478170979415846, 2.15999979740506238796366021183, 2.95440724832837064510945733095, 4.02619197900537234653869128669, 5.43562095897497227288491159121, 5.94194987005716777417466882588, 7.13290944209961654065967088343, 7.68980610564291871131544125167, 9.098566798818769925975626306144, 9.690816688234801809005562874992
