| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 4.88·5-s − 5.78·7-s + 2.82i·8-s + 6.91i·10-s − 19.4i·11-s + 6.00·13-s + 8.17i·14-s + 4.00·16-s + 14.4·17-s − 21.2i·19-s + 9.77·20-s − 27.4·22-s + 12.7·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.977·5-s − 0.825·7-s + 0.353i·8-s + 0.691i·10-s − 1.76i·11-s + 0.461·13-s + 0.583i·14-s + 0.250·16-s + 0.849·17-s − 1.11i·19-s + 0.488·20-s − 1.24·22-s + 0.552·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7326745653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7326745653\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 113 | \( 1 + (-39.5 - 105. i)T \) |
| good | 5 | \( 1 + 4.88T + 25T^{2} \) |
| 7 | \( 1 + 5.78T + 49T^{2} \) |
| 11 | \( 1 + 19.4iT - 121T^{2} \) |
| 13 | \( 1 - 6.00T + 169T^{2} \) |
| 17 | \( 1 - 14.4T + 289T^{2} \) |
| 19 | \( 1 + 21.2iT - 361T^{2} \) |
| 23 | \( 1 - 12.7T + 529T^{2} \) |
| 29 | \( 1 - 35.8T + 841T^{2} \) |
| 31 | \( 1 - 27.2T + 961T^{2} \) |
| 37 | \( 1 + 27.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.57iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.75iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 21.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 48.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 5.43T + 3.48e3T^{2} \) |
| 61 | \( 1 + 16.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8.46iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 103.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 11.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 65.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 2.76iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 158.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 160.T + 9.40e3T^{2} \) |
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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
−8.451775014249186765302153004114, −8.055046128687083853224002012560, −6.87984474270626374227961913339, −6.11583984046165032751501620388, −5.16381394549239275807861478651, −4.10068281171054228582565372483, −3.28364765469424844167197817918, −2.85880062813494549302838339529, −1.00410310663514862685270373956, −0.24558224560765469224128857973,
1.27817666478462452021145918568, 2.89216271254650244813883320502, 3.86415438603171648420175225189, 4.51621061586063831409607233849, 5.47898888650511639997594941650, 6.52333153302378497571327284491, 7.01071519350803997400021673450, 7.919954302403837444745145142201, 8.292580327878458658591754526415, 9.540475594074211077639517370223
