| L(s) = 1 | + (1.99 + 0.125i)2-s + (3.96 + 0.5i)4-s + 7i·7-s + (7.85 + 1.49i)8-s + 10.7·11-s + (−0.876 + 13.9i)14-s + (15.5 + 3.96i)16-s + (21.4 + 1.34i)22-s + 17.2·23-s − 25·25-s + (−3.5 + 27.7i)28-s − 53.1i·29-s + (30.4 + 9.86i)32-s − 63.4·37-s + 58i·43-s + (42.5 + 5.36i)44-s + ⋯ |
| L(s) = 1 | + (0.998 + 0.0626i)2-s + (0.992 + 0.125i)4-s + i·7-s + (0.982 + 0.186i)8-s + 0.974·11-s + (−0.0626 + 0.998i)14-s + (0.968 + 0.248i)16-s + (0.972 + 0.0610i)22-s + 0.748·23-s − 25-s + (−0.125 + 0.992i)28-s − 1.83i·29-s + (0.951 + 0.308i)32-s − 1.71·37-s + 1.34i·43-s + (0.967 + 0.121i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.96343 + 0.741164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.96343 + 0.741164i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 - 0.125i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 11 | \( 1 - 10.7T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 17.2T + 529T^{2} \) |
| 29 | \( 1 + 53.1iT - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 + 63.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 58iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 70.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 63.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 140.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−11.83574513626538727785473970238, −11.46703700325389534995930569372, −10.07957456303401733676618737791, −8.969021192829510757334107393223, −7.80731983666238942070187772390, −6.56084036867114395131466875089, −5.77288082574938148390766211133, −4.59833705539675066037411554006, −3.34014488768183894206219169106, −1.94551603612797745868564281348,
1.47373139088970916969720606805, 3.33839351870542647499122746856, 4.25949314480912043115892761941, 5.45944151106339210312721821663, 6.75600043648757164077069149847, 7.34773170006779796628765319946, 8.834630526335959733129697986897, 10.17078414576021223145047288117, 10.92920365403455218317665133795, 11.87152352798704381466010314116
