| L(s) = 1 | − 1.75·2-s − 1.73·3-s − 0.909·4-s − 0.711i·5-s + 3.04·6-s + 2.64i·7-s + 8.63·8-s + 2.99·9-s + 1.25i·10-s − 8.01i·11-s + 1.57·12-s − 19.1·13-s − 4.65i·14-s + 1.23i·15-s − 11.5·16-s + 25.4i·17-s + ⋯ |
| L(s) = 1 | − 0.879·2-s − 0.577·3-s − 0.227·4-s − 0.142i·5-s + 0.507·6-s + 0.377i·7-s + 1.07·8-s + 0.333·9-s + 0.125i·10-s − 0.728i·11-s + 0.131·12-s − 1.47·13-s − 0.332i·14-s + 0.0821i·15-s − 0.720·16-s + 1.49i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5156564006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5156564006\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (19.3 - 12.4i)T \) |
| good | 2 | \( 1 + 1.75T + 4T^{2} \) |
| 5 | \( 1 + 0.711iT - 25T^{2} \) |
| 11 | \( 1 + 8.01iT - 121T^{2} \) |
| 13 | \( 1 + 19.1T + 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 5.80iT - 361T^{2} \) |
| 29 | \( 1 + 13.4T + 841T^{2} \) |
| 31 | \( 1 - 49.8T + 961T^{2} \) |
| 37 | \( 1 + 61.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 69.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 8.79T + 2.20e3T^{2} \) |
| 53 | \( 1 + 63.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 54.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 65.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 14.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 74.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 70.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 68.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 12.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 135. iT - 9.40e3T^{2} \) |
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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
−10.37814776438557059344775978511, −9.851147233951182888493805595983, −8.811162458053665618558855041916, −8.113143635593011340003151412423, −7.15241699080114833356349574453, −5.92580594907668186651298283811, −5.01656258911794192522724459435, −3.87154334061172103836998576112, −2.00069637010863805525173181951, −0.43609450542163329764290622764,
0.875608767442893677633612279903, 2.58523626081237988483526087878, 4.56143472967703779699317627609, 4.91493149483042680377094936150, 6.62850653430926249105553491535, 7.34708932833543429402427886271, 8.148904648909218106329147131340, 9.494173520934191219680782519719, 9.835102682552361362811801316678, 10.65271312416562016304541129212
