L(s) = 1 | + (−133. + 44.6i)3-s − 1.42e3·5-s + 268. i·7-s + (1.56e4 − 1.18e4i)9-s − 3.57e3i·11-s − 2.44e4i·13-s + (1.88e5 − 6.34e4i)15-s + 4.61e5i·17-s − 2.81e5·19-s + (−1.19e4 − 3.57e4i)21-s + 8.13e4·23-s + 6.59e4·25-s + (−1.55e6 + 2.27e6i)27-s − 6.98e6·29-s + 3.97e6i·31-s + ⋯ |
L(s) = 1 | + (−0.948 + 0.318i)3-s − 1.01·5-s + 0.0422i·7-s + (0.797 − 0.603i)9-s − 0.0735i·11-s − 0.237i·13-s + (0.963 − 0.323i)15-s + 1.34i·17-s − 0.495·19-s + (−0.0134 − 0.0400i)21-s + 0.0606·23-s + 0.0337·25-s + (−0.564 + 0.825i)27-s − 1.83·29-s + 0.772i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.3664357840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3664357840\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (133. - 44.6i)T \) |
good | 5 | \( 1 + 1.42e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 268. iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 3.57e3iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 2.44e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 4.61e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 2.81e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.13e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.97e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 6.19e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.56e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 2.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.87e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.23e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.09e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 9.71e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 6.32e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.04e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.16e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.79e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 4.03e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 9.79e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 - 6.36e8T + 7.60e17T^{2} \) |
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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
−10.86871362142600748764572389675, −9.982162426227531388870152592774, −8.664414403789916686271260918581, −7.63472679988650365173494481007, −6.52851302850355524053227213124, −5.48766316254134087868168587415, −4.29964157454756416373032335778, −3.49292577903543837223586592309, −1.55482367824980567800946739538, −0.16859483500170302270225348007,
0.56649080803605562015794766511, 2.05334013329472018446318157823, 3.73190072910423819822984464710, 4.74435919861907501245288133674, 5.84292699373353277479403419135, 7.12157033854218688499486570760, 7.65421898688607845273143688661, 9.045550088440300320509152216730, 10.25231164014887833705585046686, 11.38943751408830244798837814991