| L(s) = 1 | + (3.80 + 1.23i)2-s + (1.59 − 2.19i)3-s + (12.9 + 9.40i)4-s + (55.0 + 9.87i)5-s + (8.79 − 6.38i)6-s − 152. i·7-s + (37.6 + 51.7i)8-s + (72.8 + 224. i)9-s + (197. + 105. i)10-s + (−60.8 + 187. i)11-s + (41.3 − 13.4i)12-s + (969. − 315. i)13-s + (188. − 580. i)14-s + (109. − 105. i)15-s + (79.1 + 243. i)16-s + (181. + 249. i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (0.102 − 0.141i)3-s + (0.404 + 0.293i)4-s + (0.984 + 0.176i)5-s + (0.0997 − 0.0724i)6-s − 1.17i·7-s + (0.207 + 0.286i)8-s + (0.299 + 0.922i)9-s + (0.623 + 0.333i)10-s + (−0.151 + 0.466i)11-s + (0.0829 − 0.0269i)12-s + (1.59 − 0.517i)13-s + (0.257 − 0.791i)14-s + (0.125 − 0.120i)15-s + (0.0772 + 0.237i)16-s + (0.151 + 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.93198 + 0.309432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.93198 + 0.309432i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.80 - 1.23i)T \) |
| 5 | \( 1 + (-55.0 - 9.87i)T \) |
| good | 3 | \( 1 + (-1.59 + 2.19i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 152. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (60.8 - 187. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-969. + 315. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-181. - 249. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (182. - 132. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (4.07e3 + 1.32e3i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (4.71e3 + 3.42e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (3.64e3 - 2.64e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.18e4 - 3.86e3i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.59e3 + 4.89e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.73e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.10e4 - 1.51e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.53e3 + 2.11e3i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (597. + 1.83e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-5.75e3 + 1.77e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.99e4 - 5.50e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (9.73e3 + 7.07e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-8.15e4 - 2.65e4i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.51e4 + 2.55e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.46e4 + 2.02e4i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.19e4 - 9.82e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.03e4 - 2.80e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.14487596874959298180130885499, −13.61324927082153684961286418726, −12.76445423696853330450093937739, −10.87700091334823485177855816937, −10.16618014713383947223766440051, −8.175763632116988518700681337392, −6.87981323193663388195245101880, −5.54526921339725092415611946433, −3.87921091150406115843076978800, −1.80654913350617930466787256331,
1.76734906005105050322783420582, 3.57424554401206452789686042168, 5.54980316509578711333754868615, 6.36496445451497095196730148323, 8.704327140228581642699171941766, 9.664743975130797345022470936394, 11.18765126785132327457422094057, 12.34032715110422664267859960780, 13.37707180812786346643408562899, 14.36832678804078754237569204400
