L(s) = 1 | + (1.40 − 1.40i)3-s + (−1.58 + 1.58i)5-s − 0.552·7-s + 5.05i·9-s + (3.85 + 3.85i)11-s + (11.5 + 11.5i)13-s + 4.44i·15-s + 26.8·17-s + (15.9 − 15.9i)19-s + (−0.775 + 0.775i)21-s − 26.3·23-s − 5.00i·25-s + (19.7 + 19.7i)27-s + (12.3 + 12.3i)29-s − 0.502i·31-s + ⋯ |
L(s) = 1 | + (0.468 − 0.468i)3-s + (−0.316 + 0.316i)5-s − 0.0788·7-s + 0.561i·9-s + (0.350 + 0.350i)11-s + (0.890 + 0.890i)13-s + 0.296i·15-s + 1.57·17-s + (0.838 − 0.838i)19-s + (−0.0369 + 0.0369i)21-s − 1.14·23-s − 0.200i·25-s + (0.731 + 0.731i)27-s + (0.424 + 0.424i)29-s − 0.0162i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.88926 + 0.332335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88926 + 0.332335i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.58 - 1.58i)T \) |
good | 3 | \( 1 + (-1.40 + 1.40i)T - 9iT^{2} \) |
| 7 | \( 1 + 0.552T + 49T^{2} \) |
| 11 | \( 1 + (-3.85 - 3.85i)T + 121iT^{2} \) |
| 13 | \( 1 + (-11.5 - 11.5i)T + 169iT^{2} \) |
| 17 | \( 1 - 26.8T + 289T^{2} \) |
| 19 | \( 1 + (-15.9 + 15.9i)T - 361iT^{2} \) |
| 23 | \( 1 + 26.3T + 529T^{2} \) |
| 29 | \( 1 + (-12.3 - 12.3i)T + 841iT^{2} \) |
| 31 | \( 1 + 0.502iT - 961T^{2} \) |
| 37 | \( 1 + (-4.19 + 4.19i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 47.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-3.50 - 3.50i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 32.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (25.8 - 25.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-62.5 - 62.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-60.5 - 60.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-31.3 + 31.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 93.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 16.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 94.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (100. - 100. i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 48.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 99.3T + 9.40e3T^{2} \) |
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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
−11.57846411556437414642296140419, −10.56189145539744553398386638520, −9.549363593794064342891388212326, −8.511338061254772779898709523725, −7.61815224692390698665851766420, −6.84685615410156102698627174660, −5.58357206500119602226921758342, −4.16836928674433233582917841829, −2.94583681546325423351984334495, −1.47792243618725948274806634542,
1.03952540104101442745446227267, 3.25149431113782996928034105044, 3.85525447449411545133760653144, 5.40771054676786201217145650872, 6.34817404685149190291904665931, 7.931003365053298176764551477816, 8.377257092386262167798966844501, 9.669292547017257164824447926416, 10.12012779859665383027039534277, 11.50010929468956848573180413225