| L(s) = 1 | − 2-s + 4-s − 2.16·5-s − 2.82·7-s − 8-s + 2.16·10-s + 4.02·11-s − 6.45·13-s + 2.82·14-s + 16-s − 1.41·17-s + 4.95·19-s − 2.16·20-s − 4.02·22-s − 23-s − 0.293·25-s + 6.45·26-s − 2.82·28-s + 0.683·29-s − 2.22·31-s − 32-s + 1.41·34-s + 6.12·35-s − 6.39·37-s − 4.95·38-s + 2.16·40-s − 7.70·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.970·5-s − 1.06·7-s − 0.353·8-s + 0.686·10-s + 1.21·11-s − 1.79·13-s + 0.754·14-s + 0.250·16-s − 0.343·17-s + 1.13·19-s − 0.485·20-s − 0.857·22-s − 0.208·23-s − 0.0587·25-s + 1.26·26-s − 0.533·28-s + 0.126·29-s − 0.399·31-s − 0.176·32-s + 0.242·34-s + 1.03·35-s − 1.05·37-s − 0.804·38-s + 0.343·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5329275008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5329275008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4.02T + 11T^{2} \) |
| 13 | \( 1 + 6.45T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 29 | \( 1 - 0.683T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 - 5.20T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 + 8.34T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.0997T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 + 3.07T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.588911085620373206966894261578, −7.70491897647369914457544731309, −7.05851822069427614436601429881, −6.71424557413330419047327849020, −5.62164286921812053369428460090, −4.65032791797171978370429392464, −3.66886477683690145460595454934, −3.07234978957888675813016396162, −1.88963490954172484795326280457, −0.45459346952605490711864933424,
0.45459346952605490711864933424, 1.88963490954172484795326280457, 3.07234978957888675813016396162, 3.66886477683690145460595454934, 4.65032791797171978370429392464, 5.62164286921812053369428460090, 6.71424557413330419047327849020, 7.05851822069427614436601429881, 7.70491897647369914457544731309, 8.588911085620373206966894261578
