L(s) = 1 | − 4.78i·2-s + 9.48·3-s − 6.93·4-s + (23.8 − 7.57i)5-s − 45.4i·6-s + (−9.59 + 48.0i)7-s − 43.4i·8-s + 9·9-s + (−36.2 − 114. i)10-s − 134.·11-s − 65.7·12-s − 54.4·13-s + (230. + 45.9i)14-s + (226. − 71.8i)15-s − 318.·16-s + 411.·17-s + ⋯ |
L(s) = 1 | − 1.19i·2-s + 1.05·3-s − 0.433·4-s + (0.953 − 0.302i)5-s − 1.26i·6-s + (−0.195 + 0.980i)7-s − 0.678i·8-s + 0.111·9-s + (−0.362 − 1.14i)10-s − 1.11·11-s − 0.456·12-s − 0.321·13-s + (1.17 + 0.234i)14-s + (1.00 − 0.319i)15-s − 1.24·16-s + 1.42·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.55393 - 1.39093i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55393 - 1.39093i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-23.8 + 7.57i)T \) |
| 7 | \( 1 + (9.59 - 48.0i)T \) |
good | 2 | \( 1 + 4.78iT - 16T^{2} \) |
| 3 | \( 1 - 9.48T + 81T^{2} \) |
| 11 | \( 1 + 134.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 54.4T + 2.85e4T^{2} \) |
| 17 | \( 1 - 411.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 561. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 30.3iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 66.2T + 7.07e5T^{2} \) |
| 31 | \( 1 + 912. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.60e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 241. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.13e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 517.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 145. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.87e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.94e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.90e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 428.T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.36e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.05e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 630.T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.30e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 61.6T + 8.85e7T^{2} \) |
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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
−15.26444276461833376847201320316, −14.04184443294313759008556984445, −12.92821884376501335316564994206, −12.03394652223611931498870687285, −10.23166893587670835012226558831, −9.480964234263274401833223966315, −8.100305964363809850557277435020, −5.66806535565878929366672364506, −3.12588661669237353532786683790, −2.01888583232201417016074087974,
2.74955382712773363476653049562, 5.34940426863180357246323810730, 6.98454700235663065169350223685, 7.979841955097090455877347005430, 9.423879485224790103219476223477, 10.76506411591968550212631752763, 13.16706663083391198954373702445, 14.04921329310922324369344556063, 14.71418767864803715044633090798, 15.96721196240684998000577427295