| L(s) = 1 | + 3-s + 5-s − 0.0644·7-s + 9-s − 3.34·11-s + 5.28·13-s + 15-s − 2.77·17-s + 1.21·19-s − 0.0644·21-s − 3.55·23-s + 25-s + 27-s + 3.07·29-s + 31-s − 3.34·33-s − 0.0644·35-s + 3.28·37-s + 5.28·39-s + 6.98·41-s − 5.78·43-s + 45-s + 11.3·47-s − 6.99·49-s − 2.77·51-s + 12.5·53-s − 3.34·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.0243·7-s + 0.333·9-s − 1.00·11-s + 1.46·13-s + 0.258·15-s − 0.673·17-s + 0.279·19-s − 0.0140·21-s − 0.742·23-s + 0.200·25-s + 0.192·27-s + 0.570·29-s + 0.179·31-s − 0.582·33-s − 0.0108·35-s + 0.539·37-s + 0.845·39-s + 1.09·41-s − 0.882·43-s + 0.149·45-s + 1.65·47-s − 0.999·49-s − 0.388·51-s + 1.72·53-s − 0.451·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.825443129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.825443129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 0.0644T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 17 | \( 1 + 2.77T + 17T^{2} \) |
| 19 | \( 1 - 1.21T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.71T + 67T^{2} \) |
| 71 | \( 1 - 9.39T + 71T^{2} \) |
| 73 | \( 1 + 5.18T + 73T^{2} \) |
| 79 | \( 1 - 8.25T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.958343902338060999663700876475, −7.34243408183270336235670966898, −6.36027394644589444431576854009, −5.96114056504486542729174566684, −5.05645448441592341807839003765, −4.26351607044107485803282515170, −3.48231305328226216981294892658, −2.65024279640150904276286541015, −1.93760138329469513593956155020, −0.829177864174630384100069462689,
0.829177864174630384100069462689, 1.93760138329469513593956155020, 2.65024279640150904276286541015, 3.48231305328226216981294892658, 4.26351607044107485803282515170, 5.05645448441592341807839003765, 5.96114056504486542729174566684, 6.36027394644589444431576854009, 7.34243408183270336235670966898, 7.958343902338060999663700876475
