| L(s) = 1 | + (0.569 + 5.16i)3-s + (8.28 + 14.3i)5-s + (−3.56 + 18.1i)7-s + (−26.3 + 5.88i)9-s + (21.0 + 12.1i)11-s + (21.6 + 12.5i)13-s + (−69.3 + 50.9i)15-s + (−38.0 − 65.8i)17-s + (22.4 + 12.9i)19-s + (−95.8 − 8.07i)21-s + (59.2 − 34.1i)23-s + (−74.7 + 129. i)25-s + (−45.4 − 132. i)27-s + (72.7 − 41.9i)29-s − 174. i·31-s + ⋯ |
| L(s) = 1 | + (0.109 + 0.993i)3-s + (0.740 + 1.28i)5-s + (−0.192 + 0.981i)7-s + (−0.975 + 0.218i)9-s + (0.575 + 0.332i)11-s + (0.462 + 0.266i)13-s + (−1.19 + 0.877i)15-s + (−0.542 − 0.940i)17-s + (0.271 + 0.156i)19-s + (−0.996 − 0.0839i)21-s + (0.536 − 0.309i)23-s + (−0.597 + 1.03i)25-s + (−0.323 − 0.946i)27-s + (0.465 − 0.268i)29-s − 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.895624983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895624983\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.569 - 5.16i)T \) |
| 7 | \( 1 + (3.56 - 18.1i)T \) |
| good | 5 | \( 1 + (-8.28 - 14.3i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-21.0 - 12.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-21.6 - 12.5i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (38.0 + 65.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-22.4 - 12.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-59.2 + 34.1i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-72.7 + 41.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 174. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-2.25 + 3.90i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (176. - 305. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-210. - 364. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 526.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (263. - 151. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 611.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 698. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 486. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (397. - 229. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 5.23T + 4.93e5T^{2} \) |
| 83 | \( 1 + (69.2 + 119. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (610. - 1.05e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-793. + 458. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.53186498341681526776446192839, −11.16288109356414184961461638279, −9.794127172688645298942579021301, −9.542179025113418949652226560853, −8.345348909424424667301663800449, −6.73339972878266293614149054188, −6.00583904990474950336644807906, −4.75865869990749749815380195460, −3.25334260497204103381868133709, −2.34294992538407236858516681785,
0.76459659655104657523549836217, 1.68320058638063271810684344914, 3.59156106948483304745347228720, 5.09701401876152726135784835201, 6.19758939849397682149347870547, 7.11983347517668420220726413678, 8.443454360439242677411799972998, 8.940985835631802842002795941219, 10.21024122136905033737491182589, 11.31128533903436120444843618984
